Tüm alıştırma soruları

What is the minimum integer value of $x$ that satisfies the inequality $|2x - 7| \le 5$ ?

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Cevap: 1

Cevap

The minimum integer value of

x

that satisfies the inequality is 1.

To solve

|2x - 7| \le 5

, write it as the compound inequality

-5 \le 2x - 7 \le 5

. Adding 7 across all parts gives

2 \le 2x \le 12

. Dividing by 2 yields

1 \le x \le 6

. The integer solutions are 1, 2, 3, 4, 5, and 6. The minimum integer among these is 1.

Adım Adım Çözüm

1

Convert the absolute value inequality into a compound inequality.

-5 \le 2x - 7 \le 5

An inequality of the form

|u| \le a

(where

a \ge 0

) is equivalent to

-a \le u \le a

.

2

Add 7 to all three parts of the inequality.

2 \le 2x \le 12

Adding a constant to an inequality preserves the direction of the inequality signs.

3

Divide all three parts by 2.

1 \le x \le 6

Dividing by a positive constant isolates

x

without reversing the inequality signs.

4

Determine the minimum integer within the solution set

[1, 6]

.

1

The solution set contains integers

\{1, 2, 3, 4, 5, 6\}

, making 1 the smallest integer value.

Anahtar Kavram

Solving absolute value inequalities using equivalent compound linear inequalities

Soru 82Soru →

If $x$ is a real number such that $|2x - 7| \le 5$ , what is the minimum possible value of $|x - 8|$ ?

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Cevap: 2

Cevap

2

Solving the given inequality

|2x - 7| \le 5

yields the compound inequality

-5 \le 2x - 7 \le 5

. Adding 7 across the inequality gives

2 \le 2x \le 12

, which simplifies to

1 \le x \le 6

. Geometrically,

|x - 8|

represents the distance between

x

and 8 on the real number line. To minimize this distance for any

x

in the closed interval

[1, 6]

, we select the point in

[1, 6]

closest to 8, which is

x = 6

. Evaluating at

x = 6

produces

|6 - 8| = 2

.

Adım Adım Çözüm

1

Unpack the absolute value inequality

1 \le x \le 6

The inequality

|2x - 7| \le 5

is equivalent to

-5 \le 2x - 7 \le 5

. Adding 7 gives

2 \le 2x \le 12

, and dividing by 2 yields

1 \le x \le 6

.

2

Determine the value in the domain

[1, 6]

that minimizes

|x - 8|

x = 6

The expression

|x - 8|

measures the distance from

x

to 8 on the number line. The value within

[1, 6]

nearest to 8 is

x = 6

.

3

Evaluate the expression at

x = 6

2

Substituting

x = 6

into

|x - 8|

gives

|6 - 8| = |-2| = 2

.

Anahtar Kavram

Properties of Linear Inequalities and Absolute Value as Distance

Tahmini Süre:1m 30s

Soru 83Soru →

The table below shows the frequency distribution of the weights, in grams, for a sample of 80 manufactured components.

Weight (grams)	Frequency
100–109	12
110–119	24
120–129	28
130–139	16

What percentage of the components in the sample have a weight of at least 120 grams?

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Cevap: 55

Cevap

55%

To calculate the percentage of components with a weight of at least 120 grams, sum the frequencies of all weight intervals corresponding to 120 grams or more:

28 + 16 = 44

. Next, divide this count by the total number of components (

80

) and multiply by

100

:

\frac{44}{80} \times 100\% = 55\%

.

Adım Adım Çözüm

1

Identify the number of components with weight at least 120 grams

44 components

The weight intervals '120–129' and '130–139' contain 28 and 16 components respectively, giving

28 + 16 = 44

.

2

Calculate the percentage relative to the total sample size of 80 components

55%

Dividing the target count (44) by the total sample size (80) and multiplying by 100 yields

\frac{44}{80} \times 100 = 55\%

.

Anahtar Kavram

Calculating class percentages from grouped frequency tables

Soru 84Soru →

In the $xy$ -plane, line $l$ passes through the points $(2, 5)$ and $(6, 13)$ . What is the slope of line $l$ ?

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Cevap: 2

Cevap

The slope of line

l

is 2.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(2, 5)

and

(6, 13)

into the formula gives

m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

.

Adım Adım Çözüm

1

Identify the coordinates of the two given points on line

l

.

The points are

(x_1, y_1) = (2, 5)

and

(x_2, y_2) = (6, 13)

.

Two points are required to calculate the slope of a straight line.

2

Apply the slope formula

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

.

m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

.

The slope measures the vertical change (rise) divided by the horizontal change (run).

Anahtar Kavram

Slope of a line through two points

Soru 85Soru →

For all real numbers $x$ satisfying the absolute value inequality $|4x - 12| \le 20$ , the maximum possible value of the expression $|2 - 3x|$ is $M$ . What is the value of $M$ ?

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Cevap: 22

Cevap

The maximum possible value

M

of the expression

|2 - 3x|

on the domain

-2 \le x \le 8

is 22.

Solving

|4x - 12| \le 20

yields

-20 \le 4x - 12 \le 20

, which simplifies to

-2 \le x \le 8

. Evaluating

|2 - 3x|

over this interval gives a minimum of

0

(at

x = 2/3

) and endpoint values of

|2 - 3(-2)| = 8

and

|2 - 3(8)| = |-22| = 22

. Thus, the maximum value

M

is 22.

Adım Adım Çözüm

1

Unfold the given absolute value inequality into a compound linear inequality.

-20 \le 4x - 12 \le 20

The inequality

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate the variable

x

by adding 12 and dividing by 4.

-2 \le x \le 8

Adding 12 gives

-8 \le 4x \le 32

. Dividing by positive 4 preserves inequality signs, yielding

-2 \le x \le 8

.

3

Evaluate the target expression

|2 - 3x|

at the boundary points of the interval

[-2, 8]

.

For

x = -2

:

|2 - 3(-2)| = |8| = 8

. For

x = 8

:

|2 - 3(8)| = |-22| = 22

.

The expression

f(x) = |2 - 3x|

is convex and non-negative, reaching its local minimum of 0 at

x = \frac{2}{3}

. Its maximum over a closed interval must occur at one of the endpoints.

4

Compare the evaluated values to find the maximum

M

.

M = \max(8, 22) = 22

Comparing 8 and 22 shows that 22 is the absolute maximum value achievable within the domain.

Anahtar Kavram

Solving linear absolute value inequalities to determine variable bounds and evaluating extreme values of absolute value expressions.

Tahmini Süre:2m 0s

Soru 86Soru →

In a right triangle, the lengths of the two legs are in a ratio of $3:4$ . If the perimeter of the triangle is $36$ centimeters, what is the area of the triangle, in square centimeters?

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Cevap: 54

Cevap

The area of the triangle is 54 square centimeters.

Since the ratio of the legs of the right triangle is 3:4, the triangle forms a classic 3-4-5 right triangle proportion. The perimeter is

3x + 4x + 5x = 12x

. Setting

12x = 36

yields

x = 3

. The legs are therefore

9

cm and

12

cm. Calculating the area using

\frac{1}{2} \times 9 \times 12

gives

54

square centimeters.

Adım Adım Çözüm

1

Express the side lengths in terms of a variable

x

Legs are

3x

and

4x

, and hypotenuse is

5x

By the Pythagorean theorem, a right triangle with legs in ratio 3:4 has hypotenuse ratio

\sqrt{3^2 + 4^2} = 5

.

2

Solve for

x

using the given perimeter

x = 3

The sum of all three sides is

3x + 4x + 5x = 12x = 36

, giving

x = 3

.

3

Calculate the actual leg lengths and area

Legs are

9

cm and

12

cm; Area is

54

\text{cm}^2

The area of a right triangle is half the product of its perpendicular legs:

\frac{1}{2} \times 9 \times 12 = 54

.

Anahtar Kavram

Perimeter and area of right triangles using standard side ratios

Soru 87Soru →

What is the sum of all integer values of $x$ that satisfy both $|2x - 5| \le 9$ and $|x + 2| \ge 4$ ?

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Cevap: 27

Cevap

The sum of all integer values of

x

that satisfy both inequalities is 27.

First, solving

|2x - 5| \le 9

yields

-9 \le 2x - 5 \le 9 \implies -4 \le 2x \le 14 \implies -2 \le x \le 7

. Second, solving

|x + 2| \ge 4

yields

x + 2 \ge 4 \implies x \ge 2

or

x + 2 \le -4 \implies x \le -6

. Taking the intersection of

-2 \le x \le 7

and

(x \ge 2 \text{ or } x \le -6)

gives the solution set

2 \le x \le 7

. The integer values satisfying this condition are 2, 3, 4, 5, 6, and 7. Summing these integers gives

2 + 3 + 4 + 5 + 6 + 7 = 27

.

Adım Adım Çözüm

1

Solve the inequality

|2x - 5| \le 9

-2 \le x \le 7

An inequality of the form

|u| \le a

(where

a \ge 0

) expands to

-a \le u \le a

. Adding 5 gives

-4 \le 2x \le 14

, and dividing by 2 yields

-2 \le x \le 7

.

2

Solve the inequality

|x + 2| \ge 4

x \ge 2 \text{ or } x \le -6

An inequality of the form

|u| \ge a

(where

a > 0

) expands to

u \ge a

or

u \le -a

. Subtracting 2 from both inequalities yields

x \ge 2

or

x \le -6

.

3

Determine the overlapping interval for both inequalities

2 \le x \le 7

Combining

-2 \le x \le 7

with

x \ge 2 \text{ or } x \le -6

eliminates

x \le -6

. The intersection of

[-2, 7]

and

[2, \infty)

is

[2, 7]

.

4

Identify the integer values in the solution interval and calculate their sum

27

The integers in the closed interval

[2, 7]

are 2, 3, 4, 5, 6, and 7. Adding them together gives

2 + 3 + 4 + 5 + 6 + 7 = 27

.

Anahtar Kavram

System of Linear Absolute Value Inequalities

Soru 88Soru →

Dataset $X$ consists of 2 numbers, each equal to 10. Dataset $Y$ consists of 8 numbers, each equal to 25. If Dataset $X$ and Dataset $Y$ are combined to form a single dataset of 10 numbers, what is the standard deviation of the combined dataset?

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Cevap: 6

Cevap

The standard deviation of the combined dataset is 6.

To find the standard deviation of the combined dataset, first calculate the combined mean:

\frac{2(10) + 8(25)}{10} = 22

. Next, compute the variance by finding the average of the squared deviations from 22:

\frac{2(10 - 22)^2 + 8(25 - 22)^2}{10} = \frac{2(144) + 8(9)}{10} = \frac{360}{10} = 36

. Taking the square root of the variance yields a standard deviation of

\sqrt{36} = 6

.

Adım Adım Çözüm

1

Calculate the mean of the combined 10-number dataset.

The combined mean is 22.

The mean of the combined dataset is needed to compute individual deviations.

2

Compute the squared deviation of each data point from the combined mean and average them to determine the variance.

The variance is 36.

Variance is defined as the arithmetic mean of the squared deviations from the mean.

3

Calculate the square root of the variance to find the standard deviation.

The standard deviation is 6.

Standard deviation is the non-negative square root of variance.

Anahtar Kavram

Standard deviation of a combined dataset

Soru 89Soru →

A company allocates a total budget of $\$4,000$ between its marketing and research departments. The amount allocated to marketing is $\$400$ more than three times the amount allocated to research. How many dollars are allocated to research?

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Cevap: 900

Cevap

The amount allocated to research is

900

dollars.

If

x

represents the research budget, the marketing budget is

3x + 400

. Summing both department allocations gives

x + (3x + 400) = 4,000

. Simplifying this linear equation gives

4x + 400 = 4,000

, leading to

4x = 3,600

and

x = 900

.

Adım Adım Çözüm

1

Define the unknown variable and express both allocations algebraically.

Let

x

be the research budget. The marketing budget is

3x + 400

.

The marketing allocation is defined relative to the research allocation.

2

Formulate a linear equation representing the combined budget.

x + (3x + 400) = 4000

The total budget allocated across both departments is

\$4,000

.

3

Solve the linear equation for

x

.

4x + 400 = 4000 \implies 4x = 3600 \implies x = 900

Combine like terms, isolate the variable term by subtracting

400

, and divide by

4

.

Anahtar Kavram

Setting up and solving a linear equation in one variable from a real-life word problem context.

Soru 90Soru →

In a circle centered at point $O$ , the ratio of the area of sector $AOB$ to the area of the entire circle is $3:8$ . If the total perimeter of sector $AOB$ is $24 + 9\pi$ , what is the radius of the circle?

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Cevap: 12

Cevap

The radius of the circle is 12.

Since the ratio of the sector area to the total circle area is

3:8

, the sector accounts for

\frac{3}{8}

of the entire circumference

2\pi r

, making the arc length

L = \frac{3}{8}(2\pi r) = \frac{3}{4}\pi r

. The perimeter of the sector is

2r + L = 2r + \frac{3}{4}\pi r

. Equating this to

24 + 9\pi

gives

2r = 24

, so

r = 12

.

Adım Adım Çözüm

1

Relate sector area ratio to arc length

Arc length

L = \frac{3}{4}\pi r

The fraction of the circle occupied by the sector is

\frac{3}{8}

, so the arc length is

\frac{3}{8}

of the circle's circumference

2\pi r

.

2

Formulate the perimeter expression for sector

AOB

Perimeter =

2r + \frac{3}{4}\pi r

The boundary of a sector includes two straight radii of length

r

plus the curved arc length

L

.

3

Equate to given perimeter and solve for

r

r = 12

Comparing

2r + \frac{3}{4}\pi r

to

24 + 9\pi

, setting

2r = 24

yields

r = 12

, which also satisfies

\frac{3}{4}\pi (12) = 9\pi

.

Anahtar Kavram

Perimeter of a circle sector and fractional relationship between sector area, central angle, and arc length.

Soru 91Soru →

If

x

is a positive real number satisfying the equation

\sqrt[3]{\frac{x^3 \sqrt{x\sqrt{x}}}{x^{-1/4}}} = 16

what is the value of

x

?

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Cevap: 8

Cevap

8

By converting all radicals into fractional exponents and systematically applying exponent rules, the expression under the cube root simplifies to

x^4

. Taking the cube root gives

x^{4/3} = 16

. Solving for

x

by raising both sides to

3/4

yields

x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

.

Adım Adım Çözüm

1

Express the inner nested radical using fractional exponents

\sqrt{x\sqrt{x}} = \sqrt{x \cdot x^{1/2}} = \sqrt{x^{3/2}} = x^{3/4}

Applying the product and power rules of exponents:

x^a \cdot x^b = x^{a+b}

and

(x^a)^b = x^{ab}

.

2

Simplify the numerator inside the outer radical

x^3 \cdot x^{3/4} = x^{3 + 3/4} = x^{15/4}

Multiplying exponential terms with the same base requires adding their exponents.

3

Divide by the negative exponent in the denominator

\frac{x^{15/4}}{x^{-1/4}} = x^{15/4 - (-1/4)} = x^{16/4} = x^4

Dividing exponential terms with the same base requires subtracting the denominator exponent from the numerator exponent.

4

Apply the outer cube root to the simplified expression

\sqrt[3]{x^4} = (x^4)^{1/3} = x^{4/3}

The

n

-th root of an expression is equivalent to raising that expression to the power of

1/n

.

5

Solve the resulting exponential equation for

x

x^{4/3} = 16 \implies x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

Raise both sides of

x^{4/3} = 16

to the power of

3/4

to isolate

x

.

Anahtar Kavram

Simplifying nested algebraic radicals and solving equations with fractional exponents using exponent rules.

Soru 92Soru →

In a geometric plane, line $L_1$ is parallel to line $L_2$ . Points $A$ and $C$ lie on line $L_1$ , and points $B$ and $D$ lie on line $L_2$ . Line segments $AB$ and $CD$ intersect at point $X$ located between lines $L_1$ and $L_2$ . If measure of $\angle XAC = 42^\circ$ and measure of $\angle XDB = 35^\circ$ , what is the measure, in degrees, of $\angle AXC$ ?

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Cevap: 103

Cevap

The measure of

\angle AXC

is

103^\circ

.

Line

L_1

is parallel to line

L_2

, meaning segment

AC

is parallel to segment

BD

. Transversal line

CD

intersects both parallel lines, creating alternate interior angles

\angle XCA

and

\angle XDB

. Hence,

\angle XCA = \angle XDB = 35^\circ

. Inside triangle

ACX

, the three interior angles must sum to

180^\circ

. Substituting the values yields

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

.

Adım Adım Çözüm

1

Identify parallel lines and the transversal line

Line segment

CD

acts as a transversal line intersecting parallel lines

L_1

and

L_2

.

Points

A

and

C

lie on line

L_1

while points

B

and

D

lie on line

L_2

with

L_1 \parallel L_2

.

2

Apply the alternate interior angles theorem

\angle XCA = \angle XDB = 35^\circ

When a transversal intersects two parallel lines, alternate interior angles are equal.

3

Calculate the target angle using the sum of interior angles in a triangle

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

The sum of interior angles in triangle

ACX

is

180^\circ

.

Anahtar Kavram

Properties of parallel lines intersected by a transversal and the triangle angle sum theorem.

Soru 93Soru →

In the $xy$ -plane, line $k$ passes through the point $(1, -2)$ and is perpendicular to line $m$ , which is defined by the equation $3x - 4y = 12$ . Line $k$ intersects line $n$ , defined by the equation $y = 2x + 1$ , at point $P$ . What is the distance between point $P$ and the point $(3.5, 3)$ ?

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Cevap: 5

Cevap

5

Rewriting the equation of line

m

,

3x - 4y = 12

, in slope-intercept form gives

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

, so line

m

has a slope of

\frac{3}{4}

. Since line

k

is perpendicular to line

m

, its slope is the negative reciprocal,

-\frac{4}{3}

. Using the point

(1, -2)

, line

k

has the equation

y = -\frac{4}{3}x - \frac{2}{3}

. Setting this equal to the equation of line

n

(

y = 2x + 1

) yields

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

, which solves to

x = -0.5

and

y = 0

, giving the intersection point

P(-0.5, 0)

. Finally, calculating the distance between

P(-0.5, 0)

and

(3.5, 3)

using the distance formula gives

\sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Adım Adım Çözüm

1

Determine the slope of line

m

The slope of line

m

is

\frac{3}{4}

Rewriting

3x - 4y = 12

in slope-intercept form gives

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

, so the slope is

\frac{3}{4}

.

2

Determine the slope of line

k

The slope of line

k

is

-\frac{4}{3}

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

3

Write the equation of line

k

The equation of line

k

is

y = -\frac{4}{3}x - \frac{2}{3}

Using point-slope form with

(1, -2)

gives

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

, which simplifies to

y = -\frac{4}{3}x - \frac{2}{3}

.

4

Find the coordinates of intersection point

P

Point

P

has coordinates

(-0.5, 0)

Setting

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

yields

-\frac{10}{3}x = \frac{5}{3} \implies x = -0.5

. Substituting

x = -0.5

into

y = 2x + 1

yields

y = 0

.

5

Calculate the distance between

P(-0.5, 0)

and

(3.5, 3)

The distance is

5

Applying the distance formula yields

d = \sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Anahtar Kavram

Perpendicular line slopes, finding intersection of two lines, and applying the distance formula.

Soru 94Soru →

If $x$ is a real number that satisfies the inequality $3|x - 4| + 2|x + 5| \le 26$ , what is the maximum possible value of $|x - 7|$ ?

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Cevap: 11

Cevap

11

Solving the piecewise linear inequality

3|x - 4| + 2|x + 5| \le 26

yields the interval

[-4, 5.6]

. The distance function

|x - 7|

reaches its maximum at the endpoint farthest from

7

, which is

x = -4

. Evaluating

|-4 - 7|

yields 11.

Adım Adım Çözüm

1

Identify the critical points of the absolute value terms.

The critical points are

x = -5

(where

x + 5 = 0

) and

x = 4

(where

x - 4 = 0

).

Critical points mark where the linear expressions inside the absolute values change sign.

2

Analyze the inequality piecewise across the three regions defined by the critical points.

For

x < -5

:

3(4 - x) + 2(-5 - x) \le 26 \implies 2 - 5x \le 26 \implies x \ge -4.8

. This produces no solution since

x

cannot be simultaneously

< -5

and

\ge -4.8

.
For

-5 \le x < 4

:

3(4 - x) + 2(x + 5) \le 26 \implies 22 - x \le 26 \implies x \ge -4

, yielding

-4 \le x < 4

.
For

x \ge 4

:

3(x - 4) + 2(x + 5) \le 26 \implies 5x - 2 \le 26 \implies x \le 5.6

, yielding

4 \le x \le 5.6

.

Expanding absolute value terms according to their regional sign definitions removes the absolute values.

3

Combine the valid regional solutions to establish the complete solution interval for

x

.

The set of all satisfying real numbers is

x \in [-4, 5.6]

.

Taking the union of the non-empty piecewise solution intervals yields the total solution set.

4

Find the maximum value of

|x - 7|

over

x \in [-4, 5.6]

.

At

x = -4

,

|-4 - 7| = |-11| = 11

. At

x = 5.6

,

|5.6 - 7| = |-1.4| = 1.4

. The maximum possible value is 11.

The expression

|x - 7|

measures distance from

7

. The maximum distance on a closed interval occurs at the endpoint farthest from

7

, which is

x = -4

.

Anahtar Kavram

Piecewise analysis of linear absolute value inequalities and optimization of absolute value distance functions.

Tahmini Süre:2m 30s

Soru 95Soru →

For all real numbers $p$ and $q$ , the custom operation $\odot$ is defined by $p \odot q = 2p^2 - 3q$ . What is the value of $3 \odot (-4)$ ?

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Cevap: 30

Cevap

The value of

3 \odot (-4)

is

30

.

By applying the definition of the custom operation

p \odot q = 2p^2 - 3q

with

p = 3

and

q = -4

, we obtain

2(3)^2 - 3(-4) = 2(9) + 12 = 18 + 12 = 30

.

Adım Adım Çözüm

1

Substitute

p = 3

and

q = -4

into the expression

2p^2 - 3q

.

2(3)^2 - 3(-4)

The custom operation defines how to process the two inputs

p

and

q

.

2

Simplify the powers and products according to order of operations.

2(9) - (-12) = 18 + 12

Exponents must be calculated before multiplication, and multiplying two negative numbers yields a positive value.

3

Add the terms together to get the final numerical result.

30

18 + 12 = 30

.

Anahtar Kavram

Custom Symbol Operations

Soru 96Soru →

In $\triangle ABC$ , the lengths of sides $AB$ , $BC$ , and $AC$ are $13$ , $14$ , and $15$ , respectively. A line segment $DE$ is drawn parallel to side $BC$ , with point $D$ lying on side $AB$ and point $E$ lying on side $AC$ . If the perimeter of $\triangle ADE$ is equal to the perimeter of quadrilateral $DBCE$ , what is the area of $\triangle ADE$ ?

Cevabı ve açıklamayı göster

Cevap: 47.25

Cevap

47.25

The area of the original triangle

\triangle ABC

is computed as

84

using Heron's formula. By defining the linear scale factor

k

between

\triangle ADE

and

\triangle ABC

, the perimeters of

\triangle ADE

and quadrilateral

DBCE

are expressed as

42k

and

42 - 14k

, respectively. Setting these equal yields

k = 0.75

. The area of

\triangle ADE

is then

k^2 \times 84 = 0.5625 \times 84 = 47.25

.

Adım Adım Çözüm

1

Calculate the perimeter and area of the main triangle

\triangle ABC

.

The perimeter of

\triangle ABC

is

13 + 14 + 15 = 42

. Using Heron's formula with semi-perimeter

s = 21

,

\text{Area}(\triangle ABC) = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84

.

Finding the area and perimeter of the full triangle sets the required baseline for proportional scaling.

2

Set up expressions for the perimeters of

\triangle ADE

and quadrilateral

DBCE

using a scale factor

k

.

Because

DE \parallel BC

,

\triangle ADE \sim \triangle ABC

with scale factor

k = \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}

. Thus,

\text{Perimeter}(\triangle ADE) = 13k + 15k + 14k = 42k

. The segments

DB = 13(1-k)

and

EC = 15(1-k)

, so

\text{Perimeter}(DBCE) = 13(1-k) + 14 + 15(1-k) + 14k = 42 - 14k

.

Parallel lines create similar triangles, which allows all perimeter segment lengths to be represented in terms of one variable

k

.

3

Solve for the scale factor

k

by equating the two perimeters.

42k = 42 - 14k \implies 56k = 42 \implies k = \frac{42}{56} = \frac{3}{4} = 0.75

.

Equating the perimeters satisfies the condition specified in the question stem.

4

Calculate the area of

\triangle ADE

using the square of the linear scale factor.

\text{Area}(\triangle ADE) = k^2 \times \text{Area}(\triangle ABC) = \left(\frac{3}{4}\right)^2 \times 84 = \frac{9}{16} \times 84 = \frac{189}{4} = 47.25

.

The area ratio of similar geometric figures is proportional to the square of their linear scale factor.

Anahtar Kavram

Properties of Similar Triangles, Area via Heron's Formula, and Perimeter Scaling

Soru 97Soru →

A water reservoir is filled by Pipe A and Pipe B operating simultaneously at their respective constant rates. Operating together at their original rates, the two pipes can fill the empty reservoir completely in $12$ hours. On a certain day, both pipes begin filling the empty reservoir together at their original rates. After $4$ hours, Pipe A's rate decreases by $25\%$ , while Pipe B's rate increases by $50\%$ . Operating at these new constant rates, the two pipes require an additional $7$ hours to fill the remainder of the reservoir. How many hours would it take Pipe A, operating alone at its original rate, to fill the entire reservoir?

Cevabı ve açıklamayı göster

Cevap: 25.2

Cevap

It would take Pipe A 25.2 hours operating alone at its original rate to fill the entire reservoir.

By defining the original work rates

a

and

b

in reservoirs per hour, the initial condition yields

a + b = \frac{1}{12}

. In the first 4 hours,

\frac{1}{3}

of the job is completed, leaving

\frac{2}{3}

. Setting up the equation for the remaining job with modified rates

0.75a

and

1.5b

over 7 hours produces

7(0.75a + 1.5b) = \frac{2}{3}

. Solving this system of two linear equations yields

a = \frac{5}{126}

reservoirs per hour. Taking the reciprocal gives the time required for Pipe A alone to fill the reservoir, which is

25.2

hours.

Adım Adım Çözüm

1

Set up equations for the original rates of Pipe A (

a

) and Pipe B (

b

).

The combined original rate is

a + b = \frac{1}{12}

reservoir per hour.

Together they complete

1

reservoir in

12

hours.

2

Determine the fraction of the reservoir filled in the first 4 hours and the remaining fraction.

Work completed =

4 \times \frac{1}{12} = \frac{1}{3}

; Remaining work =

\frac{2}{3}

.

The pipes worked at their original combined rate for 4 hours.

3

Set up an equation for the work done during the remaining 7 hours at the adjusted rates.

7 \left(0.75a + 1.5b\right) = \frac{2}{3} \implies 5.25a + 10.5b = \frac{2}{3} \implies 63a + 126b = 8

.

Pipe A's rate decreases by

25\%

to

0.75a

, and Pipe B's rate increases by

50\%

to

1.5b

.

4

Solve the system of linear equations for

a

.

a = \frac{5}{126}

reservoir per hour.

Multiplying

a + b = \frac{1}{12}

by

126

yields

126a + 126b = 10.5

. Subtracting

63a + 126b = 8

gives

63a = 2.5

, so

a = \frac{2.5}{63} = \frac{5}{126}

.

5

Calculate the time for Pipe A alone to fill the entire reservoir.

Time

= \frac{1}{a} = \frac{126}{5} = 25.2

hours.

Time equals total work divided by individual rate.

Anahtar Kavram

Algebraic modeling of combined work and rates with mid-process rate modifications

Soru 98Soru →

In circle $O$ , line segments $AB$ and $CD$ are perpendicular diameters, each of length $12$ . An arc of a second circle, centered at point $A$ with radius $AC$ , is drawn from point $C$ to point $D$ through the interior of circle $O$ . What is the area of the crescent-shaped region bounded by the semicircle $CBD$ of circle $O$ and arc $CD$ of the second circle?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

Circle

O

has radius

r = 6

, giving semicircle

CBD

an area of

\frac{1}{2}\pi(6^2) = 18\pi

. The distance

AC = \sqrt{6^2 + 6^2} = 6\sqrt{2}

is the radius of circle

A

. Because

\angle CAD = 90^\circ

, sector

ACD

has area

\frac{90^\circ}{360^\circ}\pi(6\sqrt{2})^2 = 18\pi

. Subtracting the area of triangle

ACD

(

\frac{1}{2} \times 12 \times 6 = 36

) yields a segment area of

18\pi - 36

. Subtracting this segment area from the semicircle area yields

18\pi - (18\pi - 36) = 36

.

Adım Adım Çözüm

1

Find the radii of circle

O

and circle

A

.

Radius of circle

O

is

r = 6

. In right triangle

AOC

,

OA = OC = 6

, so radius

AC = \sqrt{6^2 + 6^2} = 6\sqrt{2}

.

Perpendicular diameters

AB

and

CD

intersect at center

O

, dividing each diameter into radii of length

6

.

2

Calculate the area of sector

ACD

of circle

A

and triangle

ACD

.

Sector area

= \frac{90^\circ}{360^\circ} \pi (6\sqrt{2})^2 = 18\pi

. Triangle area

= \frac{1}{2} \times 12 \times 6 = 36

.

Angle

\angle CAD = 90^\circ

because

\triangle ACD

is a right isosceles triangle with hypotenuse

CD = 12

.

3

Find the area of the circular segment bounded by chord

CD

and arc

CD

of circle

A

.

Segment Area

= 18\pi - 36

.

The area of a circular segment is equal to the sector area minus the triangle area.

4

Subtract the segment area from the area of semicircle

CBD

of circle

O

.

Region Area

= 18\pi - (18\pi - 36) = 36

.

Semicircle

CBD

has radius

6

and area

\frac{1}{2}\pi(6^2) = 18\pi

. Subtracting the segment area leaves the crescent region.

Anahtar Kavram

Area of circular sectors, segments, and compound regions (Lune of Hippocrates)

Soru 99Soru →

A customer service representative resolved the following number of support tickets over five consecutive days: $18$ , $24$ , $15$ , $31$ , and $22$ . What is the arithmetic mean of the number of tickets resolved per day by the representative?

Cevabı ve açıklamayı göster

Cevap: 22

Cevap

The arithmetic mean of the number of tickets resolved per day is

22

.

The mean of a data set is calculated by taking the sum of all values and dividing by the total number of items. The sum of the numbers is

18 + 24 + 15 + 31 + 22 = 110

. Dividing

110

by

5

yields

22

.

Adım Adım Çözüm

1

Sum all data values in the set

18 + 24 + 15 + 31 + 22 = 110

To find the mean, the first step is to calculate the total sum of all observations.

2

Divide the total sum by the total number of values

\frac{110}{5} = 22

The arithmetic mean is defined as the sum of the values divided by the count of the values.

Anahtar Kavram

Arithmetic Mean

Soru 100Soru →

An electronics manufacturer produces three types of circuit boards: Alpha, Beta, and Gamma. Production requires processing across three specialized workstations: Solder, Component Placement, and Inspection.

- Each Alpha board requires 2 hours of Solder, 3 hours of Component Placement, and 1 hour of Inspection.
- Each Beta board requires 1 hour of Solder, 4 hours of Component Placement, and 2 hours of Inspection.
- Each Gamma board requires 3 hours of Solder, 2 hours of Component Placement, and 4 hours of Inspection.

During a given production cycle, the Solder station was operated for 55 hours, the Component Placement station for 85 hours, and the Inspection station for 65 hours. If all three workstations were operated at full capacity with no downtime, what was the total number of circuit boards produced?

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

The total number of circuit boards produced is 28.

Setting up equations for total machine hours yields

2x + y + 3z = 55

,

3x + 4y + 2z = 85

, and

x + 2y + 4z = 65

, where

x

,

y

, and

z

represent the quantities of Alpha, Beta, and Gamma boards produced, respectively. Subtracting the third equation from the first equation gives

(2x + y + 3z) - (x + 2y + 4z) = 55 - 65

, which simplifies to

x - y - z = -10

, or

x = y + z - 10

. Substituting

x = y + z - 10

into the second and third equations produces the 2x2 system

7y + 5z = 115

and

3y + 5z = 75

. Subtracting these two equations eliminates

z

, giving

4y = 40

, so

y = 10

. Substituting

y = 10

into

3y + 5z = 75

gives

30 + 5z = 75

, so

z = 9

. Substituting

y = 10

and

z = 9

into

x = y + z - 10

gives

x = 9

. The total number of circuit boards produced is

x + y + z = 9 + 10 + 9 = 28

.

Adım Adım Çözüm

1

Set up the 3x3 system of linear equations based on workstation hours.

2x + y + 3z = 55

,

3x + 4y + 2z = 85

, and

x + 2y + 4z = 65

Each equation models the total operational hours used across the three product types.

2

Subtract the third equation from the first equation to isolate

x

in terms of

y

and

z

.

x = y + z - 10

Eliminating terms directly reduces coefficient complexity.

3

Substitute

x = y + z - 10

into the second and third equations to construct a 2x2 system.

3y + 5z = 75

and

7y + 5z = 115

Reducing to a two-variable system allows direct elimination.

4

Subtract the two reduced equations to solve for

y

and

z

.

y = 10

and

z = 9

The

5z

terms cancel out upon subtraction.

5

Determine

x

and compute the total sum

x + y + z

.

x = 9

, total

= 9 + 10 + 9 = 28

The question asks for the total quantity of circuit boards produced.

Anahtar Kavram

Systems of Linear Equations

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