Tüm alıştırma soruları

Soru 101Soru →

A $10$ -foot straight ladder leans against a vertical wall on flat horizontal ground. If the bottom of the ladder is $6$ feet away from the base of the wall, how high up the wall, in feet, does the top of the ladder reach?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The top of the ladder reaches

8

feet up the wall.

Applying the Pythagorean theorem

a^2 + b^2 = c^2

with hypotenuse

c = 10

and ground leg

a = 6

gives

6^2 + b^2 = 10^2

, which simplifies to

36 + b^2 = 100

, so

b^2 = 64

and

b = 8

feet.

Adım Adım Çözüm

1

Set up the right triangle relationship using the Pythagorean theorem.

a^2 + b^2 = c^2

, where

a = 6

and

c = 10

.

The ladder forms the hypotenuse of a right triangle with the vertical wall and horizontal ground.

2

Substitute the known values into the formula and solve for

b

.

6^2 + b^2 = 10^2 \implies 36 + b^2 = 100 \implies b^2 = 64 \implies b = 8

.

Subtracting

36

from

100

gives

64

, and taking the positive square root yields the vertical height.

Anahtar Kavram

Applying the Pythagorean Theorem (

a^2 + b^2 = c^2

) to find an unknown leg length in a right triangle context.

Soru 102Soru →

For all real numbers

x

such that

x \neq 0

and

x \neq 1

, the function

f

satisfies the equation

f(x) + f\left(\frac{x - 1}{x}\right) = 1 + x

What is the value of

f(2)

?

Cevabı ve açıklamayı göster

Cevap: 0.75

Cevap

The value of

f(2)

is

0.75

(or

\frac{3}{4}

).

Evaluating the given equation at

x = 2

,

x = 1/2

, and

x = -1

generates a system of three linear equations:

f(2) + f(1/2) = 3

,

f(1/2) + f(-1) = 1.5

, and

f(-1) + f(2) = 0

. Adding all three equations yields

2(f(2) + f(1/2) + f(-1)) = 4.5

, which simplifies to

f(2) + f(1/2) + f(-1) = 2.25

. Subtracting the equation

f(1/2) + f(-1) = 1.5

from

2.25

gives

f(2) = 0.75

.

Adım Adım Çözüm

1

Determine the cyclic sequence of inputs starting at

x = 2

Evaluating the inner transformation

g(x) = \frac{x-1}{x}

yields

g(2) = \frac{1}{2}

,

g\left(\frac{1}{2}\right) = -1

, and

g(-1) = 2

. The input sequence cycles every 3 steps.

Identifying that

g(g(g(x))) = x

allows constructing a closed system of linear equations.

2

Set up a system of three linear equations using the cycle values

For

x = 2

:

f(2) + f\left(\frac{1}{2}\right) = 3

.
For

x = \frac{1}{2}

:

f\left(\frac{1}{2}\right) + f(-1) = \frac{3}{2}

.
For

x = -1

:

f(-1) + f(2) = 0

.

Substituting each element of the cycle into the given relation produces three linear equations in three unknown function values.

3

Sum the equations and isolate

f(2)

Summing all three equations gives

2\left[f(2) + f\left(\frac{1}{2}\right) + f(-1)\right] = \frac{9}{2}

, so

f(2) + f\left(\frac{1}{2}\right) + f(-1) = \frac{9}{4}

. Subtracting

f\left(\frac{1}{2}\right) + f(-1) = \frac{3}{2}

yields

f(2) = \frac{9}{4} - \frac{6}{4} = \frac{3}{4} = 0.75

.

Symmetric summation provides the total sum of the three function values, allowing direct elimination of the other two variables.

Anahtar Kavram

Solving cyclic functional equations by forming a system of linear equations

Soru 103Soru →

A non-degenerate triangle has integer side lengths $a$ , $b$ , and $c$ such that $a \le b \le c$ . If the perimeter of the triangle is $30$ and its area is an integer, what is the length of the longest side $c$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

Using Heron's formula with semi-perimeter

s = 15

, the area of the triangle is

\sqrt{15(15-a)(15-b)(15-c)}

. By defining

x = 15-a

,

y = 15-b

, and

z = 15-c

, we have

x+y+z = 15

with

x \ge y \ge z \ge 1

. For the area to be an integer,

15xyz

must be a perfect square. The AM-GM inequality bounds

xyz \le 125

, leaving

xyz = 60

as the only valid case (

k=2

). The system

x+y+z=15

and

xyz=60

has a unique positive integer solution

(10, 3, 2)

, giving side lengths

(5, 12, 13)

. Thus, the longest side length

c

is

13

.

Adım Adım Çözüm

1

Find the semi-perimeter of the triangle.

s = \frac{a+b+c}{2} = \frac{30}{2} = 15

Heron's formula requires the semi-perimeter

s

to compute the area of a triangle.

2

Set up Heron's formula for the area.

\text{Area} = \sqrt{15(15-a)(15-b)(15-c)}

Heron's formula relates the side lengths of a triangle directly to its area.

3

Define variables for the differences from the semi-perimeter.

Let

x = 15-a

,

y = 15-b

, and

z = 15-c

. Then

x+y+z = 15

and

x \ge y \ge z \ge 1

.

Because

a \le b \le c

, their complements with respect to

15

satisfy

x \ge y \ge z

.

4

Determine the condition for the area to be an integer.

\text{Area} = \sqrt{15xyz}

must be an integer, so

15xyz = m^2

for some integer

m

, implying

xyz = 15k^2

for an integer

k \ge 1

.

For

\sqrt{15xyz}

to be an integer,

15xyz

must be a perfect square containing

15

as a factor.

5

Bound

xyz

using the AM-GM inequality.

xyz \le \left(\frac{x+y+z}{3}\right)^3 = 5^3 = 125 \implies 15k^2 \le 125 \implies k=1 \text{ or } k=2

The AM-GM inequality provides a maximum upper bound for the product of three positive numbers with a fixed sum.

6

Evaluate the possible values of

k

.

If

k=1

,

xyz = 15

, which has no integer solutions for

x+y+z=15

. If

k=2

,

xyz = 60

, which yields

(x, y, z) = (10, 3, 2)

.

Solving the system

x+y+z=15

and

xyz=60

under

x \ge y \ge z \ge 1

gives

z=2

,

y=3

,

x=10

uniquely.

7

Calculate the side lengths and identify the longest side

c

.

a = 5

,

b = 12

,

c = 13

, so the longest side length is

13

.

Since

c = 15 - z = 15 - 2 = 13

, the longest side of the right triangle

(5, 12, 13)

is

13

.

Anahtar Kavram

Heron's formula, integer partitioning, and the AM-GM inequality for triangle area optimization

Soru 104Soru →

In a meteorological study, the daily peak wind speeds (in kilometers per hour) recorded over 7 consecutive days were $14, 18, 21, 24, 27, 31,$ and $37$ . As part of a data calibration process, each recorded wind speed is increased by $15\%$ and then increased by an additional constant of $5\text{ km/h}$ . What is the range, in kilometers per hour, of the calibrated daily peak wind speeds?

Cevabı ve açıklamayı göster

Cevap: 26.45

Cevap

The range of the calibrated daily peak wind speeds is 26.45 km/h.

For any dataset transformed linearly according to

y = a \cdot x + b

where

a > 0

, the range of the transformed dataset is given by

\text{Range}(Y) = a \cdot \text{Range}(X)

. Here, increasing each value by

15\%

corresponds to

a = 1 + 0.15 = 1.15

, and adding

5

corresponds to

b = 5

. The original maximum is

37

and the original minimum is

14

, giving an original range of

37 - 14 = 23

. Therefore, the new range is

1.15 \times 23 = 26.45\text{ km/h}

. The constant shift of

5\text{ km/h}

affects the center (mean, median) but has zero effect on the dispersion.

Adım Adım Çözüm

1

Determine the range of the original set of wind speeds.

Original Range = 37 - 14 = 23 km/h.

The range is defined as the difference between the maximum and minimum values in a dataset.

2

Apply the properties of linear transformations to measures of dispersion.

The transformation is y = 1.15x + 5. The range is scaled by 1.15 and unaffected by the addition of 5.

Adding a constant shift to all data points shifts the entire distribution without changing the spread (dispersion), whereas multiplying all data points by a constant factor 'a' scales all measures of dispersion by |a|.

3

Compute the calibrated range.

Calibrated Range = 1.15 * 23 = 26.45 km/h.

Multiplying the original range of 23 by the scale factor of 1.15 gives the exact range of the transformed dataset.

Anahtar Kavram

Effect of linear transformations on measures of dispersion (range, standard deviation, IQR)

Tahmini Süre:1m 30s

Soru 105Soru →

If $x$ is an integer that satisfies both $|3x - 12| \le 15$ and $|2x + 4| \ge 6$ , what is the sum of the maximum possible value of $x$ and the minimum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The correct answer is 10.

Solving the first inequality

|3x - 12| \le 15

yields

-1 \le x \le 9

. Solving the second inequality

|2x + 4| \ge 6

yields

x \ge 1

or

x \le -5

. The values of

x

that satisfy both constraints are those in the interval

[1, 9]

. Within this interval, the minimum integer value is

1

and the maximum integer value is

9

. Adding these values together gives

1 + 9 = 10

.

Adım Adım Çözüm

1

Solve the bounded absolute value inequality

|3x - 12| \le 15

.

-15 \le 3x - 12 \le 15 \implies -3 \le 3x \le 27 \implies -1 \le x \le 9.

An inequality of the form

|A| \le B

(where

B \ge 0

) unwraps to the compound inequality

-B \le A \le B

.

2

Solve the boundary absolute value inequality

|2x + 4| \ge 6

.

2x + 4 \ge 6 \implies 2x \ge 2 \implies x \ge 1

, or

2x + 4 \le -6 \implies 2x \le -10 \implies x \le -5$.

An inequality of the form

|A| \ge B

(where

B > 0

) splits into two separate cases:

A \ge B

or

A \le -B

.

3

Find the intersection of the solution sets from Step 1 and Step 2.

The intersection of

[-1, 9]

and

(-\infty, -5] \cup [1, \infty)

is

[1, 9]

.

To satisfy both inequalities simultaneously,

x

must belong to both set

[-1, 9]

and set

(-\infty, -5] \cup [1, \infty)

.

4

Identify the extreme integer values of

x

and compute their sum.

\text{Minimum integer } x = 1, \quad \text{Maximum integer } x = 9; \quad 1 + 9 = 10.

The range of valid integer values for

x

is all integers from 1 to 9 inclusive.

Anahtar Kavram

Linear Inequalities and Absolute Value

Tahmini Süre:2m 0s

Soru 106Soru →

A company has two sales divisions, Division X and Division Y. Division X has 12 representatives with an average monthly sales volume of $\$14,000$ . Division Y has 18 representatives with an average monthly sales volume of $\$24,000$ . If 2 representatives leave Division Y who together accounted for $\$82,000$ in monthly sales, what is the new combined average monthly sales volume, in dollars, for all remaining representatives in both divisions?

Cevabı ve açıklamayı göster

Cevap: 18500

Cevap

The new combined average monthly sales volume for all remaining representatives is $18,500.

To find the combined mean, calculate the sum of all remaining values across both groups (

518,000

) and divide by the total number of remaining items (

28

), yielding

18,500

.

Adım Adım Çözüm

1

Find total monthly sales for Division X

12 \times 14,000 = 168,000

The total value of a group is the product of its count and its arithmetic mean.

2

Find total monthly sales for Division Y before representatives left

18 \times 24,000 = 432,000

Multiply the number of representatives in Division Y by their average sales volume.

3

Determine the remaining total sales in Division Y

432,000 - 82,000 = 350,000

Subtract the sales volume of the 2 departing representatives from Division Y's original total.

4

Calculate the total combined sales for both divisions

168,000 + 350,000 = 518,000

Add the total sales of Division X and the remaining sales of Division Y.

5

Calculate total number of remaining representatives across both divisions

12 + (18 - 2) = 28

Division X retains 12 representatives, while Division Y retains 16 representatives.

6

Compute the combined mean sales volume

\frac{518,000}{28} = 18,500

Divide the total combined sales volume by the total number of remaining representatives.

Anahtar Kavram

Weighted Mean and Combining Group Data

Soru 107Soru →

In isosceles triangle $ABC$ , side $AB$ is equal in length to side $AC$ . The perimeter of triangle $ABC$ is $36$ , and the length of the altitude from vertex $A$ to base $BC$ is $12$ . What is the area of triangle $ABC$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The area of triangle

ABC

is

60

.

Let

x

be the length of the two equal sides

AB

and

AC

, and

b

be the length of base

BC

. The perimeter is

2x + b = 36

, yielding

x = 18 - \frac{b}{2}

. The altitude from

A

to

BC

has length

12

and bisects

BC

into two segments of length

\frac{b}{2}

. Applying the Pythagorean theorem to one of the right triangles gives

x^2 = 12^2 + \left(\frac{b}{2}\right)^2

. Substituting

x = 18 - \frac{b}{2}

gives

\left(18 - \frac{b}{2}\right)^2 = 144 + \frac{b^2}{4} \implies 324 - 18b + \frac{b^2}{4} = 144 + \frac{b^2}{4} \implies 18b = 180 \implies b = 10

. The area is

\frac{1}{2} \times 10 \times 12 = 60

.

Adım Adım Çözüm

1

Set up an equation for the side lengths using the perimeter.

Let

b

be the length of base

BC

, and

x

be the length of sides

AB

and

AC

. Since the perimeter is

36

,

2x + b = 36

, which gives

x = 18 - \frac{b}{2}

.

An isosceles triangle has two sides of equal length, and perimeter is the sum of all three side lengths.

2

Apply the Pythagorean theorem to the right triangle formed by the altitude.

The altitude of length

12

drops perpendicularly to base

BC

, bisecting it into two equal segments of length

\frac{b}{2}

. Thus,

x^2 = 12^2 + \left(\frac{b}{2}\right)^2 = 144 + \frac{b^2}{4}

.

In an isosceles triangle, the altitude to the base bisects the base and creates two congruent right-angled triangles.

3

Solve for the base length

b

.

Substitute

x = 18 - \frac{b}{2}

into the equation:

\left(18 - \frac{b}{2}\right)^2 = 144 + \frac{b^2}{4} \implies 324 - 18b + \frac{b^2}{4} = 144 + \frac{b^2}{4} \implies 18b = 180 \implies b = 10

.

Expanding the squared binomial allows the

\frac{b^2}{4}

terms to cancel out, resulting in a linear equation for

b

.

4

Calculate the area of the triangle.

\text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60.

The area of a triangle is evaluated using half the product of its base and corresponding altitude.

Anahtar Kavram

Isosceles triangle properties, altitude-to-base bisector property, Pythagorean theorem, and triangle area calculation.

Soru 108Soru →

If

x

and

y

are real numbers such that

x > y > 0

and they satisfy the following system of equations:

\frac{3}{x+y} + \frac{4}{x-y} = \frac{11}{4}

\frac{5}{x+y} - \frac{2}{x-y} = \frac{1}{4}

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is 3.

Substituting

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

transforms the non-linear looking equations into the linear system

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

. Solving this system yields

u = \frac{1}{4}

and

v = \frac{1}{2}

. Consequently,

x + y = 4

and

x - y = 2

. Adding these two equations gives

2x = 6

, so

x = 3

.

Adım Adım Çözüm

1

Introduce auxiliary variables to linearize the system.

Let

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

. The system becomes

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

.

Replacing non-linear reciprocal terms with simple variables allows elimination or substitution methods for linear systems.

2

Solve the system of linear equations for

u

and

v

using elimination.

Multiply

5u - 2v = \frac{1}{4}

by 2 to get

10u - 4v = \frac{1}{2}

. Add this to

3u + 4v = \frac{11}{4}

:

13u = \frac{11}{4} + \frac{2}{4} = \frac{13}{4} \implies u = \frac{1}{4}

. Then

4v = \frac{11}{4} - 3\left(\frac{1}{4}\right) = 2 \implies v = \frac{1}{2}

.

Eliminating

v

yields a single equation in

u

, which provides the values of both auxiliary variables.

3

Convert auxiliary values back to equations in

x

and

y

.

Since

u = \frac{1}{x+y} = \frac{1}{4}

, we get

x + y = 4

. Since

v = \frac{1}{x-y} = \frac{1}{2}

, we get

x - y = 2

.

Inverting the fractions restores the original variables in a standard 2x2 linear system.

4

Solve for

x

by adding the two linear equations.

(x + y) + (x - y) = 4 + 2 \implies 2x = 6 \implies x = 3

.

Adding the equations eliminates

y

directly, isolating

x

.

Anahtar Kavram

Solving systems of linear equations using substitution variables for algebraic simplification

Soru 109Soru →

For all real numbers $x$ , what is the numerical value of the expression $\frac{3^{x+2} - 3^x}{3^{x-1} + 3^{x-2}}$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The numerical value of the expression is 18.

Factoring

3^x

from both terms in the numerator gives

3^x(3^2 - 1) = 8 \cdot 3^x

. Factoring

3^x

from both terms in the denominator gives

3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = \frac{4}{9} \cdot 3^x

. Dividing the two expressions cancels

3^x

entirely, resulting in

\frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

.

Adım Adım Çözüm

1

Factor out

3^x

from the numerator.

3^{x+2} - 3^x = 3^x(3^2 - 1) = 3^x(9 - 1) = 8 \cdot 3^x

Applying the exponent rule

a^{m+n} = a^m \cdot a^n

allows factoring out the common exponential factor

3^x

.

2

Factor out

3^x

from the denominator.

3^{x-1} + 3^{x-2} = 3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = 3^x\left(\frac{4}{9}\right)

Applying negative exponent rules

a^{-n} = \frac{1}{a^n}

allows evaluating the remaining numerical terms inside the parentheses.

3

Simplify the overall fraction by dividing the factored numerator by the factored denominator.

\frac{8 \cdot 3^x}{\frac{4}{9} \cdot 3^x} = \frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

The non-zero common term

3^x

cancels from both numerator and denominator, leaving a constant integer.

Anahtar Kavram

Factoring and simplifying exponential expressions with variable exponents.

Soru 110Soru →

In the $xy$ -plane, line $m$ passes through the point $(4, -2)$ and has a $y$ -intercept of $6$ . What is the slope of line $m$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The slope of line

m

is

-2

.

The

y

-intercept of

6

indicates that the line passes through

(0, 6)

. Substituting

(0, 6)

and

(4, -2)

into the slope formula yields

m = \frac{6 - (-2)}{0 - 4} = \frac{8}{-4} = -2

.

Adım Adım Çözüm

1

Identify the coordinates of two points on line

m

.

The line passes through

(4, -2)

and the

y

-intercept point

(0, 6)

.

The

y

-intercept is the point where

x = 0

.

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

Substitute

(x_1, y_1) = (4, -2)

and

(x_2, y_2) = (0, 6)

into the formula:

m = \frac{6 - (-2)}{0 - 4}

.

Slope represents the ratio of vertical change to horizontal change between two points on a line.

3

Simplify the fractional expression.

m = \frac{8}{-4} = -2

.

Dividing positive

8

by negative

4

gives

-2

.

Anahtar Kavram

Slope of a line given two points or a point and intercept

Soru 111Soru →

A department consisting of $5$ employees has a mean monthly sales total of $\$12{}000$ . If a new employee with a monthly sales total of $\$18{}000$ joins the department, what is the new mean monthly sales total, in dollars, for the $6$ employees?

Cevabı ve açıklamayı göster

Cevap: 13000

Cevap

13000

To find the new mean, multiply the initial mean by the initial number of employees to get total sales:

5 \times 12{}000 = 60{}000

. Add the new employee's sales:

60{}000 + 18{}000 = 78{}000

. Finally, divide by the new total number of employees (

6

) to get

\frac{78{}000}{6} = 13{}000

.

Adım Adım Çözüm

1

Find the total sales of the original 5 employees.

5 \times 12{}000 = 60{}000

The sum of values is equal to the mean multiplied by the number of observations.

2

Calculate the total sales for all 6 employees.

60{}000 + 18{}000 = 78{}000

Add the new employee's sales to the initial total.

3

Calculate the new mean sales per employee.

\frac{78{}000}{6} = 13{}000

Divide the combined total sales by the new total number of employees (6).

Anahtar Kavram

Mean of Combined Data Sets

Tahmini Süre:1m 0s

Soru 112Soru →

A dataset consists of $25$ distinct integers arranged in increasing order. The mean of all $25$ integers is $52$ . The mean of the smallest $12$ integers is $30$ , and the mean of the largest $12$ integers is $70$ .

If $5$ additional numbers, each equal to the median of the original dataset, are added to the dataset, what is the mean of the new set of $30$ numbers?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The mean of the new set of 30 numbers is 60.

The sum of all 25 numbers is

25 \times 52 = 1300

. Since the numbers are ordered, the median is the 13th value. The 12 smallest numbers sum to

12 \times 30 = 360

and the 12 largest sum to

12 \times 70 = 840

. The sum of these 24 numbers is

360 + 840 = 1200

, so the 13th number (the median) must be

1300 - 1200 = 100

. Adding 5 copies of 100 increases the sum to

1300 + 500 = 1800

across

30

numbers. The new mean is

\frac{1800}{30} = 60

.

Adım Adım Çözüm

1

Calculate the total sum of the original dataset of 25 numbers.

Sum =

25 \times 52 = 1300

.

The mean multiplied by the number of elements gives the total sum.

2

Calculate the combined sum of the 12 smallest and 12 largest integers.

Sum of 12 smallest =

12 \times 30 = 360

; Sum of 12 largest =

12 \times 70 = 840

; Total =

360 + 840 = 1200

.

The 25 numbers consist of the 12 smallest, the 1 median (13th element), and the 12 largest.

3

Determine the value of the median.

Median =

1300 - 1200 = 100

.

Subtracting the sum of the 24 non-median values from the total sum yields the 13th element, which is the median.

4

Find the sum and count of the modified dataset.

New Sum =

1300 + 5(100) = 1800

; New Count =

25 + 5 = 30

.

Adding 5 numbers each equal to 100 increases the sum by 500 and the count by 5.

5

Calculate the mean of the new dataset.

New Mean =

\frac{1800}{30} = 60

.

Divide the new total sum by the new total count of numbers.

Anahtar Kavram

Relationship between Mean, Median, and Total Sum in Partitioned Datasets

Tahmini Süre:2m 30s

Soru 113Soru →

In the $xy$ -plane, line $L$ is given by the equation $3x + 2y = 18$ . Line $N$ is perpendicular to line $L$ and passes through the point $(-1, 4)$ . If line $L$ and line $N$ intersect at the point $(p, q)$ , what is the value of $p + q$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Converting the given equation

3x + 2y = 18

into slope-intercept form yields

y = -\frac{3}{2}x + 9

, so line

L

has slope

-\frac{3}{2}

. A perpendicular line must have a slope equal to the negative reciprocal, which is

\frac{2}{3}

. Using point

(-1, 4)

in the point-slope form gives

y - 4 = \frac{2}{3}(x + 1)

, simplifying to

2x - 3y = -14

. Solving the system of equations formed by line

L

(

3x + 2y = 18

) and line

N

(

2x - 3y = -14

) via elimination yields

x = 2

and

y = 6

. Therefore, the intersection point is

(2, 6)

, and

p + q = 2 + 6 = 8

.

Adım Adım Çözüm

1

Determine the slope of line

L

Slope of line

L

is

m_L = -\frac{3}{2}

Convert

3x + 2y = 18

to

y = -\frac{3}{2}x + 9

to identify the slope coefficient of

x

.

2

Calculate the perpendicular slope for line

N

Slope of line

N

is

m_N = \frac{2}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Derive the equation for line

N

Equation of line

N

is

2x - 3y = -14

Apply point-slope form

y - 4 = \frac{2}{3}(x + 1)

and rearrange into standard linear form.

4

Solve the system of equations to find the intersection point

(p, q)

p = 2

and

q = 6

, giving point

(2, 6)

Eliminate variable

y

by adding

3 \times (3x + 2y = 18)

and

2 \times (2x - 3y = -14)

to get

13x = 26

.

5

Sum the coordinates

p

and

q

p + q = 8

Evaluate

2 + 6 = 8

as required by the stem.

Anahtar Kavram

Perpendicular line slope relationships and linear system intersection

Tahmini Süre:2m 0s

Soru 114Soru →

An executive chartered a private aircraft to complete a trip between two cities separated by a non-stop distance of $600$ miles. On the return flight along the exact same route, strong headwinds reduced the aircraft's average ground speed by $50$ miles per hour compared to its outbound speed. As a result, the return flight took $2$ hours longer than the outbound flight. What was the average speed, in miles per hour, of the aircraft on the outbound flight?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

150 miles per hour

By setting the outbound speed to

v

and return speed to

v - 50

, the relationship between outbound flight time

\frac{600}{v}

and return flight time

\frac{600}{v - 50}

yields the rational equation

\frac{600}{v - 50} - \frac{600}{v} = 2

. Solving this equation gives the quadratic

v^2 - 50v - 15,000 = 0

, which factors as

(v - 150)(v + 100) = 0

. The positive root gives an outbound speed of 150 miles per hour.

Adım Adım Çözüm

1

Define variables for the unknown outbound rate and express travel times for both legs.

Let

v

be the outbound speed in miles per hour. Outbound time is

\frac{600}{v}

hours and return time is

\frac{600}{v - 50}

hours.

Distance equals speed multiplied by time (

d = v \cdot t

), so time equals distance divided by speed.

2

Formulate the algebraic equation using the given difference in flight durations.

\frac{600}{v - 50} - \frac{600}{v} = 2

The return flight took

2

hours longer than the outbound flight.

3

Clear denominators and simplify into standard quadratic form.

v^2 - 50v - 15,000 = 0

Multiplying both sides by

v(v - 50)

yields

600v - 600v + 30,000 = 2(v^2 - 50v)

, which simplifies to

2v^2 - 100v - 30,000 = 0

or

v^2 - 50v - 15,000 = 0

.

4

Solve the quadratic equation for

v

.

v = 150

Factoring

(v - 150)(v + 100) = 0

gives solutions

v = 150

or

v = -100

. Physical speed must be positive.

Anahtar Kavram

Distance-Rate-Time Quadratic Algebraic Modeling

Soru 115Soru →

In the $xy$ -plane, triangle $PQR$ has vertices at $P(0, 0)$ , $Q(8, 0)$ , and $R(2, 6)$ . Point $S$ lies on segment $PQ$ such that segment $RS$ divides triangle $PQR$ into two regions of equal area. Point $T$ lies on segment $QR$ such that segment $ST$ is parallel to segment $PR$ . What is the area of triangle $QST$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The area of triangle

PQR

is calculated using base

PQ = 8

and height

h = 6

, yielding

\frac{1}{2} \times 8 \times 6 = 24

. Since segment

RS

divides triangle

PQR

into two regions of equal area that share the height from vertex

R

, point

S

must be the midpoint of

PQ

, making

QS = 4

. Because segment

ST

is parallel to segment

PR

, triangle

QST

is similar to triangle

QPR

with a side length ratio of

\frac{QS}{QP} = \frac{4}{8} = \frac{1}{2}

. The ratio of the areas of similar triangles is the square of the side ratio,

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

. Multiplying the total area

24

by

\frac{1}{4}

gives the area of triangle

QST

as

6

.

Adım Adım Çözüm

1

Calculate the area of the main triangle

PQR

.

\text{Area}(PQR) = \frac{1}{2} \times 8 \times 6 = 24

.

The base

PQ

lies along the x-axis with length

8 - 0 = 8

, and the perpendicular height from vertex

R(2,6)

to the base is

6

.

2

Determine the length of segment

QS

.

QS = 4

.

Line segment

RS

splits

\triangle PQR

into two smaller triangles,

\triangle PSR

and

\triangle QSR

, which share the same altitude from vertex

R

. For their areas to be equal, their base lengths

PS

and

SQ

must be equal. Therefore,

S

is the midpoint of

PQ

, giving

QS = \frac{8}{2} = 4

.

3

Establish the similarity relationship and scale factor between

\triangle QST

and

\triangle QPR

.

\triangle QST \sim \triangle QPR

with scale factor

k = \frac{1}{2}

.

Because segment

ST

is parallel to segment

PR

, corresponding angles are equal (

\angle QST = \angle QPR

and

\angle QTS = \angle QRP

). Thus,

\triangle QST

is similar to

\triangle QPR

. The ratio of corresponding side lengths is

\frac{QS}{QP} = \frac{4}{8} = \frac{1}{2}

.

4

Compute the area of triangle

QST

.

\text{Area}(QST) = 6

.

The ratio of the areas of similar triangles is equal to the square of their linear scale factor:

\text{Area}(QST) = \left(\frac{1}{2}\right)^2 \times \text{Area}(PQR) = \frac{1}{4} \times 24 = 6

.

Anahtar Kavram

Area of triangles, midpoint area partitioning, and area ratio scaling in similar triangles

Tahmini Süre:2m 0s

Soru 116Soru →

If $x$ is a real number that satisfies $|3 - 2x| \le 9$ , what is the maximum possible value of $|x - 5|$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The maximum possible value of

|x - 5|

is 8.

Solving

|3 - 2x| \le 9

yields

-9 \le 3 - 2x \le 9

. Subtracting 3 gives

-12 \le -2x \le 6

. Dividing by

-2

and reversing inequality signs results in

-3 \le x \le 6

. To maximize

|x - 5|

, find the point in

[-3, 6]

that is farthest from 5. Evaluating at the endpoints gives

|-3 - 5| = 8

and

|6 - 5| = 1

. The maximum possible value is 8.

Adım Adım Çözüm

1

Solve the absolute value inequality

|3 - 2x| \le 9

.

-9 \le 3 - 2x \le 9

By definition of absolute value,

|u| \le k

means

-k \le u \le k

for

k \ge 0

.

2

Isolate

-2x

by subtracting 3 from each part.

-12 \le -2x \le 6

Subtracting a constant from all parts preserves the inequality direction.

3

Isolate

x

by dividing each part by

-2

.

-3 \le x \le 6

Dividing an inequality by a negative number flips the inequality signs.

4

Determine the maximum value of

|x - 5|

over the interval

[-3, 6]

.

Maximum occurs at

x = -3

, where

|-3 - 5| = 8

.

The expression

|x - 5|

represents the distance from

x

to 5 on the number line. The endpoint

x = -3

is furthest from 5.

Anahtar Kavram

Linear inequalities and absolute value as distance on a number line

Soru 117Soru →

The quadratic equation $x^2 - px + q = 0$ , where $p$ and $q$ are real constants with $p > 0$ , has two real roots $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 29$ and $r_1^3 + r_2^3 = 117$ , what is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: -10

Cevap

The value of

q

is

-10

.

By Vieta's formulas,

r_1 + r_2 = p

and

r_1 r_2 = q

. Using the identity

r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = 29

, we get

q = \frac{p^2 - 29}{2}

. Using the sum of cubes identity

r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) = 117

, we obtain

p(29 - q) = 117

. Substituting

q

into this equation yields

p^3 - 87p + 234 = 0

. Factoring this cubic equation yields

p = 3

as a valid root, leading to

q = -10

. Evaluating the discriminant

p^2 - 4q = 9 - 4(-10) = 49 > 0

confirms that real roots exist. The second positive root for

p

gives a negative discriminant, making

-10

the unique correct value of

q

.

Adım Adım Çözüm

1

Express symmetric sums of roots in terms of equation coefficients

p

and

q

using Vieta's formulas.

r_1 + r_2 = p

and

r_1 r_2 = q

. Therefore,

r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = p^2 - 2q = 29

, which gives

q = \frac{p^2 - 29}{2}

.

Relating root sums and products directly to coefficients simplifies the algebraic system.

2

Express the sum of cubes in terms of

p

and

q

and substitute

q

.

r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) = p(29 - q) = 117

. Substituting

q = \frac{p^2 - 29}{2}

yields

p\left(29 - \frac{p^2 - 29}{2}\right) = 117 \implies p(87 - p^2) = 234 \implies p^3 - 87p + 234 = 0

.

Combining symmetric expressions forms a single polynomial equation in variable

p

.

3

Factor the cubic equation

p^3 - 87p + 234 = 0

to find candidate values for

p > 0

.

Testing factor

p = 3

gives

3^3 - 87(3) + 234 = 27 - 261 + 234 = 0

. Factoring out

(p - 3)

gives

(p - 3)(p^2 + 3p - 78) = 0

.

Finding the roots of the polynomial identifies candidate values for coefficient

p

.

4

Verify the real root constraint

p^2 - 4q > 0

for candidates of

p

.

For

p = 3

,

q = \frac{3^2 - 29}{2} = -10

, giving discriminant

p^2 - 4q = 9 - 4(-10) = 49 > 0

(real roots

5

and

-2

). For the positive root

p = \frac{\sqrt{321}-3}{2} \approx 7.458

of

p^2 + 3p - 78 = 0

, the discriminant is

p^2 - 4q = 3p - 29 \approx -6.63 < 0

, which yields non-real complex roots.

The question specifies real roots, so candidate solutions resulting in negative discriminants must be discarded.

Anahtar Kavram

Quadratic Equations and Factoring (Vieta's Formulas, Symmetric Polynomials, and Real Root Conditions)

Soru 118Soru →

The running times, in minutes, for a fleet of delivery vehicles have a 25th percentile of $134$ minutes and a 75th percentile of $242$ minutes. To analyze the data in hours, a manager converts each vehicle's running time to hours by dividing by $60$ and then adds a fixed delay of $1.25$ hours to every converted value. What is the interquartile range, in hours, of the adjusted running times?

Cevabı ve açıklamayı göster

Cevap: 1.8

Cevap

The interquartile range of the adjusted running times is 1.8 hours.

The interquartile range (IQR) measures the spread of the middle 50% of a dataset (

Q_3 - Q_1

). For the original dataset in minutes,

IQR = 242 - 134 = 108

minutes. When data is transformed linearly via

Y = aX + b

, scaling all values by a positive constant

a

scales the IQR by

a

, whereas adding a constant

b

shifts all values equally and does not change the distance between percentiles. Here

a = \frac{1}{60}

and

b = 1.25

. Therefore, the new IQR in hours is

108 \times \frac{1}{60} = 1.8

hours.

Adım Adım Çözüm

1

Find the original interquartile range (IQR) in minutes

IQR = 242 - 134 = 108 minutes

The interquartile range is defined as the difference between the 75th percentile (Q3) and the 25th percentile (Q1).

2

Apply the scaling factor for unit conversion

108 / 60 = 1.8 hours

Dividing every value by 60 scales the data by a factor of 1/60, which scales measures of dispersion like IQR by |1/60|.

3

Evaluate the effect of adding the constant shift

Adjusted IQR remains 1.8 hours

Adding a constant shift to every data point shifts the entire distribution without changing the spread or distance between percentiles.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru 119Soru →

Three adjacent faces of a right rectangular prism have surface areas of $24$ , $36$ , and $54$ square centimeters. What is the volume of the prism, in cubic centimeters?

Cevabı ve açıklamayı göster

Cevap: 216

Cevap

The volume of the rectangular prism is

216

cubic centimeters.

Let the edge lengths of the right rectangular prism be

l

,

w

, and

h

. The areas of three adjacent faces meeting at a common vertex are given by

lw = 24

,

wh = 36

, and

lh = 54

. Multiplying these three equations together yields

(lw)(wh)(lh) = (lwh)^2 = 24 \times 36 \times 54 = 46,656

. Since the volume

V

of a right rectangular prism is equal to

lwh

, taking the positive square root gives

V = \sqrt{46,656} = 216

cubic centimeters.

Adım Adım Çözüm

1

Define variables for the three dimensions of the rectangular prism.

Let

l

,

w

, and

h

represent the length, width, and height of the prism.

The area of each rectangular face is the product of two adjacent dimensions.

2

Express the given face areas in terms of the dimensions.

lw = 24

,

wh = 36

, and

lh = 54

.

Adjacent faces share pairwise dimensions.

3

Multiply all three face area equations together.

(lw)(wh)(lh) = l^2 w^2 h^2 = (lwh)^2 = 24 \times 36 \times 54 = 46,656

.

Multiplying the three face areas yields the square of the prism's volume (

V = lwh

).

4

Calculate the square root of the product to determine the volume.

V = lwh = \sqrt{46,656} = 216

.

Taking the square root gives the volume directly without needing to solve for individual edge lengths.

Anahtar Kavram

For a right rectangular prism with adjacent face areas

A_1

,

A_2

, and

A_3

, the volume is given by

V = \sqrt{A_1 \times A_2 \times A_3}

.

Tahmini Süre:1m 15s

Soru 120Soru →

For all real numbers $x$ , the function $f$ is defined by $f(x) = x^2 - 4x + 7$ and the function $g$ is defined by $g(x) = 2x + 1$ . If $f(g(k)) = 19$ and $k < 0$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: -1.5

Cevap

The value of

k

is

-1.5

.

To solve for

k

, substitute

g(k) = 2k + 1

into

f(x)

to get

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

. Expanding and combining like terms yields

4k^2 - 4k + 4

. Setting this equal to

19

gives

4k^2 - 4k - 15 = 0

. Factoring the quadratic expression yields

(2k - 5)(2k + 3) = 0

, which gives

k = \frac{5}{2} = 2.5

or

k = -\frac{3}{2} = -1.5

. Since the problem specifies that

k < 0

, the value of

k

must be

-1.5

.

Adım Adım Çözüm

1

Express the nested composite function

f(g(k))

in terms of

k

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

Substituting

g(k) = 2k + 1

for every occurrence of

x

in

f(x)

sets up the composite function evaluation.

2

Expand terms and simplify the algebraic expression

f(g(k)) = 4k^2 - 4k + 4

Expanding

(2k + 1)^2 = 4k^2 + 4k + 1

and distributing

-4(2k + 1) = -8k - 4

, then combining like terms yields

4k^2 - 4k + 4

.

3

Set the composite function expression equal to 19 and form a standard quadratic equation

4k^2 - 4k - 15 = 0

Subtracting

19

from both sides produces a standard quadratic equation

ax^2 + bx + c = 0

.

4

Solve the quadratic equation and apply the condition

k < 0

k = -1.5

Factoring

(2k - 5)(2k + 3) = 0

gives

k = 2.5

or

k = -1.5

. The constraint

k < 0

selects

k = -1.5

.

Anahtar Kavram

Composite function evaluation

f(g(x))

combined with quadratic equation solving under domain constraints.

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram