Tüm alıştırma soruları

Soru 221Soru →

A local gym charges a one-time registration fee plus a constant monthly fee. The total cost, $C$ , in dollars, for a membership of $m$ months is modeled by the equation $C = 35m + 50$ . What is the one-time registration fee, in dollars, for the gym?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The one-time registration fee is 50 dollars.

In the linear model

C = 35m + 50

, the constant term

50

represents the value of the function when

m = 0

. In this context,

m = 0

corresponds to

0

months of membership, meaning no monthly fees have been incurred yet. Therefore, the value of

50

represents the initial, one-time registration fee.

Adım Adım Çözüm

1

Analyze the linear equation

C = 35m + 50

to identify the slope and the y-intercept.

The slope is

35

(the coefficient of

m

) and the y-intercept is

50

(the constant term).

In a linear equation of the form

y = mx + b

, the constant

b

represents the value of

y

when

x = 0

(the y-intercept), and the coefficient

m

represents the rate of change (the slope).

2

Interpret the meaning of the y-intercept in the given context.

At

m = 0

months, the total cost

C

is

C = 35(0) + 50 = 50

dollars.

The cost at

0

months represents the upfront, one-time fee before any monthly fees are added, which is the registration fee.

Anahtar Kavram

Interpreting the y-intercept of a linear function in a real-world context.

Soru 222Soru →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of

k

is

12

.

The quadratic equation

x^2 - kx + 36 = 0

has exactly one real solution when its discriminant is equal to zero. The discriminant of

ax^2 + bx + c = 0

is

b^2 - 4ac

. For this equation,

a = 1

,

b = -k

, and

c = 36

. Setting the discriminant to zero gives

(-k)^2 - 4(1)(36) = 0

, which simplifies to

k^2 - 144 = 0

. Solving for

k

gives

k^2 = 144

, so

k = 12

or

k = -12

. Since the problem states that

k

is a positive constant, the correct value of

k

is

12

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic equation

x^2 - kx + 36 = 0

.

a = 1

,

b = -k

, and

c = 36

.

To apply the discriminant formula, we need the values of

a

,

b

, and

c

.

2

Set the discriminant

\Delta = b^2 - 4ac

to

0

.

(-k)^2 - 4(1)(36) = 0

.

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the equation for

k

.

k^2 - 144 = 0 \implies k^2 = 144 \implies k = 12

(since

k

is a positive constant).

Solving the equation yields

k = \pm 12

, and we select the positive value as specified in the problem.

Anahtar Kavram

Discriminant of a quadratic equation

Soru 223Soru →

A parabola in the $xy$ -plane has its vertex at $(3, 8)$ and passes through the point $(5, 16)$ . If the equation of the parabola is written in the form $y = a(x - 3)^2 + 8$ , what is the value of the constant $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

Substituting the coordinates

(5, 16)

into

y = a(x - 3)^2 + 8

yields

16 = a(5 - 3)^2 + 8

. Simplifying the term

(5 - 3)^2

gives

4

, leading to the equation

16 = 4a + 8

. Subtracting 8 from both sides gives

8 = 4a

, and dividing by 4 gives the final answer of 2.

Adım Adım Çözüm

1

Substitute the point

(5, 16)

into the equation.

16 = a(5 - 3)^2 + 8

Since the parabola passes through the point

(5, 16)

, these coordinates must satisfy the equation.

2

Simplify the squared term.

16 = 4a + 8

Subtract 3 from 5 to get 2, and square 2 to get 4.

3

Solve for the constant

a

.

a = 2

Subtract 8 from both sides to get

8 = 4a

, then divide by 4 to find

a

.

Anahtar Kavram

Determining a constant in a quadratic function given its vertex form and a point on the graph.

Soru 224Soru →

An online service provider offers two monthly subscription plans. Under Plan A, the customer pays a flat monthly fee of $C$ dollars. Under Plan B, the monthly cost, in dollars, is determined by the expression $1.5(20 - x) - 0.8(3x - 5)$ , where $x$ is the number of premium features the customer uses. The provider wants Plan B to be strictly cheaper than Plan A for any customer who uses more than 4 premium features. If $C$ is an integer, what is the minimum possible value of $C$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

To find the minimum integer value of

C

, we simplify Plan B's cost expression to

34 - 3.9x

and set up the inequality

34 - 3.9x < C

. Solving for

x

by dividing by

-3.9

and reversing the inequality sign gives

x > \frac{34 - C}{3.9}

. For Plan B to be cheaper than Plan A for all customers using more than 4 features, the solution set

x > \frac{34 - C}{3.9}

must contain the interval

x > 4

. This requires the boundary point to be at most 4, so

\frac{34 - C}{3.9} \le 4

. Solving this inequality yields

C \ge 18.4

. The smallest integer value greater than or equal to

18.4

is 19.

Adım Adım Çözüm

1

Simplify the cost expression for Plan B

34 - 3.9x

To combine like terms and express Plan B's cost in standard linear form.

2

Set up the inequality stating Plan B is strictly cheaper than Plan A

34 - 3.9x < C

Plan B is cheaper than Plan A when its cost is less than

C

dollars.

3

Solve the inequality for

x

in terms of

C

x > \frac{34 - C}{3.9}

Isolating

x

allows us to find the threshold number of premium features, remembering to reverse the inequality direction when dividing by the negative coefficient

-3.9

.

4

Relate the threshold condition to the given minimum number of premium features

\frac{34 - C}{3.9} \le 4

For Plan B to be cheaper for any

x > 4

, the solution interval

x > \frac{34 - C}{3.9}

must cover the entire interval

x > 4

. Thus, the boundary point must be at most 4.

5

Solve the boundary inequality for

C

C \ge 18.4

Multiplying by

3.9

and isolating

C

gives the lower bound for the cost of Plan A.

6

Find the minimum integer value for

C

19

Since

C

must be an integer and at least

18.4

, the smallest integer that satisfies this inequality is 19.

Anahtar Kavram

Solving linear inequalities in one variable with parameter constraints and real-world conditions.

Soru 225Soru →

If $25^a = 125^b$ , where $a$ and $b$ are positive constants, what is the value of $\frac{a}{b}$ ?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

1.5

To find the value of

\frac{a}{b}

, we can rewrite the bases of the equation

25^a = 125^b

using a common base of 5:

(5^2)^a = (5^3)^b

. Applying the exponent power rule gives

5^{2a} = 5^{3b}

. Since the bases are identical, their exponents must be equal, so

2a = 3b

. Dividing both sides of this equation by

2b

yields the ratio

\frac{a}{b} = \frac{3}{2}

, which is

1.5

.

Adım Adım Çözüm

1

Express the bases as powers of 5

(5^2)^a = (5^3)^b

To solve the equation, express both sides with a common base of 5 since

25 = 5^2

and

125 = 5^3

.

2

Apply the power rule of exponents

5^{2a} = 5^{3b}

According to the exponent rules,

(x^m)^n = x^{mn}

.

3

Set the exponents equal to each other

2a = 3b

Because the bases on both sides of the equation are equal and positive, their exponents must also be equal.

4

Solve for the ratio

\frac{a}{b} = 1.5

Divide both sides of the equation

2a = 3b

by

2b

to isolate the ratio

\frac{a}{b}

.

Anahtar Kavram

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Soru 226Soru →

In the quadratic equation $8x^2 - kx + 27 = 0$ , where $k$ is a positive constant, one of the roots is the square of the other root. What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

For the quadratic equation

8x^2 - kx + 27 = 0

, Vieta's formulas state that the product of the roots is

\frac{27}{8}

and the sum of the roots is

\frac{k}{8}

. Letting the roots be

r

and

r^2

, we obtain the equation

r^3 = \frac{27}{8}

, which gives

r = \frac{3}{2}

. The other root is

r^2 = \frac{9}{4}

. Summing these roots gives

\frac{3}{2} + \frac{9}{4} = \frac{15}{4}

. Equating this sum to the formula for the sum of the roots yields

\frac{k}{8} = \frac{15}{4}

, which simplifies to

k = 30

.

Adım Adım Çözüm

1

Determine the product of the roots using Vieta's formulas.

The product of the roots is

r_1 r_2 = \frac{27}{8}

.

For any quadratic equation

ax^2 + bx + c = 0

, the product of its roots is given by

\frac{c}{a}

.

2

Use the root relationship to solve for one of the roots.

r_2^3 = \frac{27}{8} \implies r_2 = \frac{3}{2}

.

Substituting

r_1 = r_2^2

into the product equation yields

r_2^3 = \frac{27}{8}

.

3

Determine the value of the second root.

r_1 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

.

The first root is the square of the second root.

4

Calculate the sum of the two roots.

r_1 + r_2 = \frac{9}{4} + \frac{3}{2} = \frac{15}{4}

.

The sum of the roots is needed to relate back to the linear coefficient of the quadratic equation.

5

Solve for the constant

k

using the sum of the roots.

\frac{k}{8} = \frac{15}{4} \implies k = 30

.

By Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{8}

.

Anahtar Kavram

Quadratic Equations and Vieta's Formulas

Soru 227Soru →

An athlete participates in a charity event where they earn money by swimming laps in a pool and running laps on a track. The athlete earns $s$ dollars for each lap swum and $r$ dollars for each lap run. Completing 10 laps in the pool and 15 laps on the track raises a total of $135. Completing 12 laps in the pool and 20 laps on the track raises a total of$ 176. What is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

r

is 7.

The correct answer is 7. To find this, we translate the word problem into a system of two linear equations:

10s + 15r = 135

and

12s + 20r = 176

. Simplifying these equations by dividing by their common factors yields

2s + 3r = 27

and

3s + 5r = 44

. Multiplying the first equation by 3 gives

6s + 9r = 81

, and multiplying the second equation by 2 gives

6s + 10r = 88

. Subtracting the two equations eliminates

s

and gives

r = 7

.

Adım Adım Çözüm

1

Write the system of linear equations representing the athlete's earnings.

The system is:

10s + 15r = 135

12s + 20r = 176

To represent the relationships between the number of laps completed and the total amount raised.

2

Simplify both equations to work with smaller coefficients.

Divide the first equation by 5:

2s + 3r = 27

Divide the second equation by 4:

3s + 5r = 44

Simplifying equations reduces the likelihood of arithmetic errors.

3

Eliminate the variable

s

to solve for

r

.

Multiply the first simplified equation by 3 and the second by 2:

3(2s + 3r) = 3(27) \implies 6s + 9r = 81

2(3s + 5r) = 2(44) \implies 6s + 10r = 88

Subtract the first equation from the second:

(6s + 10r) - (6s + 9r) = 88 - 81 \implies r = 7

Using the elimination method allows us to solve directly for the target variable,

r

.

Anahtar Kavram

Solving a system of linear equations in two variables using elimination or substitution.

Soru 228Soru →

The system of equations below consists of a linear equation and a quadratic equation:

\begin{aligned} y - 2x &= 5 \\ y &= x^2 - 3x - 1 \end{aligned}

Let

(x_1, y_1)

and

(x_2, y_2)

represent the two distinct real coordinate points where the graphs of these equations intersect. If

y_1 > y_2

, what is the value of

x_1 - x_2

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Solving the system algebraically by expressing

y

as

2x + 5

and substituting it into the quadratic equation leads to

x^2 - 5x - 6 = 0

. Factoring this equation yields

x = 6

and

x = -1

. Evaluating these values in the linear equation gives the coordinates

(6, 17)

and

(-1, 3)

. Since

17 > 3

, the point

(6, 17)

corresponds to

(x_1, y_1)

and

(-1, 3)

corresponds to

(x_2, y_2)

. The difference

x_1 - x_2

is equal to

6 - (-1) = 7

.

Adım Adım Çözüm

1

Solve for

y

in the linear equation to express it in terms of

x

.

y = 2x + 5

This allows for substitution into the quadratic equation to eliminate one variable.

2

Substitute

2x + 5

for

y

in the quadratic equation.

2x + 5 = x^2 - 3x - 1

To create a single-variable quadratic equation in terms of

x

.

3

Rearrange the equation by subtracting

2x

and

5

from both sides to set it equal to zero.

x^2 - 5x - 6 = 0

Setting the quadratic equation to zero is the standard first step to find its roots.

4

Factor the quadratic trinomial.

(x - 6)(x + 1) = 0

, which gives

x = 6

and

x = -1

.

Factoring finds the

x

-coordinates of the points of intersection.

5

Substitute the

x

-values back into the linear equation

y = 2x + 5

to determine the corresponding

y

-values.

For

x = 6

,

y = 2(6) + 5 = 17

. For

x = -1

,

y = 2(-1) + 5 = 3

. The intersection points are

(6, 17)

and

(-1, 3)

.

This defines the full coordinate pairs of the system's solutions.

6

Apply the condition

y_1 > y_2

to identify

(x_1, y_1)

and

(x_2, y_2)

.

Since

17 > 3

, the first solution is

(x_1, y_1) = (6, 17)

and the second solution is

(x_2, y_2) = (-1, 3)

.

This aligns the coordinate values with the variables defined in the problem constraint.

7

Calculate the value of

x_1 - x_2

.

6 - (-1) = 7

Subtracting a negative value is equivalent to adding its absolute value, yielding the final required answer.

Anahtar Kavram

Nonlinear Systems of Equations

Soru 229Soru →

In the system of equations below,

k

is a constant.

\begin{aligned} x^2 + y^2 + 8y &= 9 \\ x^2 + y &= k \end{aligned}

If the system has exactly 3 distinct real solutions

(x, y)

, what is the value of

k

?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

k

is 1.

Substituting

x^2 = k - y

into

x^2 + y^2 + 8y = 9

gives

y^2 + 7y + k - 9 = 0

. For the system to have exactly 3 distinct real solutions, the vertex of the parabola

y = -x^2 + k

must lie on the circle, which corresponds to the root

y = k

. Substituting

y = k

into the quadratic equation yields

k^2 + 8k - 9 = 0

, which gives

k = 1

or

k = -9

. For

k = 1

, the roots of the quadratic are

y = 1

and

y = -8

. The root

y = 1

yields 1 real solution,

(0, 1)

, and the root

y = -8

yields 2 real solutions,

(3, -8)

and

(-3, -8)

, for a total of 3 real solutions. For

k = -9

, the roots are

y = -9

and

y = 2

. The root

y = 2

does not yield any real solutions for

x

because

2 > -9

, so the system has only 1 real solution. Thus,

k = 1

.

Adım Adım Çözüm

1

Express

x^2

in terms of

y

and

k

from the second equation.

x^2 = k - y

(with constraint

y \le k

for real

x

)

To prepare for substitution into the first equation and establish the boundary condition for real solutions.

2

Substitute

x^2 = k - y

into the first equation.

y^2 + 7y + k - 9 = 0

To create a single quadratic equation in terms of

y

.

3

Analyze the conditions on the roots of the quadratic equation to get exactly 3 distinct real solutions.

One root must equal

k

and the other must be less than

k

.

A root

y = k

yields 1 real solution for

x

(

x = 0

), while a root

y < k

yields 2 real solutions (

x = \pm\sqrt{k-y}

).

4

Find the candidate values of

k

by setting

y = k

in the quadratic equation.

k^2 + 8k - 9 = 0 \implies k = 1

or

k = -9

To find the values of

k

where the parabola's vertex lies on the circle.

5

Verify which candidate value of

k

satisfies all conditions.

For

k = 1

, the roots are

y = 1

and

y = -8 < 1

(3 solutions). For

k = -9

, the roots are

y = -9

and

y = 2 > -9

(1 solution). Therefore,

k = 1

.

To ensure the second root is strictly less than

k

, guaranteeing exactly 3 solutions.

Anahtar Kavram

Analyzing the number of solutions in a nonlinear system of equations using algebraic substitution and boundary constraints.

Tahmini Süre:3m 0s

Soru 230Soru →

An online retailer uses a linear relationship to determine the shipping cost of an order based on the total weight of the items in the order. The table below shows the shipping cost, $y$ , in dollars, for an order with a total weight of $x$ pounds.

Weight ( $x$ pounds)	Shipping Cost ( $y$ dollars)
$2$	$9.50$
$6$	$17.50$

What is the shipping cost, in dollars, for an order with a total weight of $11$ pounds?

Cevabı ve açıklamayı göster

Cevap: 27.5

Cevap

The correct shipping cost for an order weighing 11 pounds is 27.5 dollars.

The linear relationship has a constant rate of change (slope). Using the coordinates from the table,

(2, 9.50)

and

(6, 17.50)

, the slope

m

is calculated as

\frac{17.50 - 9.50}{6 - 2} = \frac{8}{4} = 2

. Using the point-slope form, the equation of the line is

y - 9.50 = 2(x - 2)

, which simplifies to

y = 2x + 5.50

. Substituting

x = 11

gives

y = 2(11) + 5.50 = 27.5

.

Adım Adım Çözüm

1

Calculate the slope (rate of change) of the linear relationship.

m = 2

Determines the rate at which the shipping cost increases per additional pound of weight.

2

Formulate the linear equation representing the relationship.

y = 2x + 5.50

Establishes a functional relationship to calculate costs for any given weight.

3

Evaluate the function at the target weight of 11 pounds.

y = 27.5

Determines the final cost of shipping for the specified weight of 11 pounds.

Anahtar Kavram

Determining and evaluating linear functions represented in tabular form

Tahmini Süre:1m 30s

Soru 231Soru →

The function $f$ is defined by $f(x) = a \cdot 3^x$ , where $a$ is a constant. If $f(2) = 45$ , what is the value of $f(1)$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the value of

f(1)

, we first determine the value of the constant

a

. We are given that

f(2) = 45

, so substituting

x = 2

into the function definition

f(x) = a \cdot 3^x

gives

45 = a \cdot 3^2

. Simplifying

3^2

to

9

yields

45 = 9a

, which means

a = 5

. Now that we know

a = 5

, we can write the function as

f(x) = 5 \cdot 3^x

. To find

f(1)

, we substitute

x = 1

into this equation, yielding

f(1) = 5 \cdot 3^1 = 15

.

Adım Adım Çözüm

1

Substitute the given point

(2, 45)

into the function equation to solve for

a

.

a = 5

Since

f(2) = 45

, we have

45 = a \cdot 3^2 = 9a

, which gives

a = 5

.

2

Evaluate the function at

x = 1

using the value of

a = 5

.

f(1) = 15

Substituting

a = 5

and

x = 1

into

f(x) = a \cdot 3^x

gives

f(1) = 5 \cdot 3^1 = 15

.

Anahtar Kavram

Evaluating and solving exponential functions given initial conditions or points.

Soru 232Soru →

A deep-sea research submersible's internal cabin pressure, $P$ , in atmospheres ( $\text{atm}$ ), is modeled by a linear function of its depth below the ocean surface, $d$ , in meters. At the surface ( $d = 0$ ), the internal pressure is $1.0\text{ atm}$ . For every increase in depth of $100\text{ meters}$ , the internal pressure increases by $0.05\text{ atm}$ . The submersible descends from the surface at a constant rate of $2.5\text{ meters per second}$ . What is the rate of increase of the internal cabin pressure, in $\text{atm}$ per hour, as the submersible descends?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

The rate of increase of the internal cabin pressure is

4.5

atmospheres per hour.

The rate of change of internal pressure with depth is

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

. Since the submersible descends at a rate of

2.5\text{ m/s}

, we convert this rate of depth change to hours:

2.5\text{ m/s} \times 3600\text{ s/hr} = 9000\text{ m/hr}

. Multiplying the pressure change per meter by the depth change per hour gives the rate of change of the internal pressure per hour:

0.0005\text{ atm/m} \times 9000\text{ m/hr} = 4.5\text{ atm/hr}

.

Adım Adım Çözüm

1

Calculate the rate of pressure increase per meter of depth.

0.0005\text{ atm/m}

The pressure increases by

0.05\text{ atm}

for every

100\text{ meters}

of depth, giving a rate of change of

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

.

2

Determine the distance descended by the submersible in one hour.

9000\text{ meters}

With

3600\text{ seconds}

in one hour and a descent speed of

2.5\text{ m/s}

, the submersible descends a total of

2.5\text{ m/s} \times 3600\text{ s} = 9000\text{ meters}

in one hour.

3

Calculate the rate of internal pressure increase per hour.

4.5\text{ atm/hr}

Multiply the rate of change of pressure per meter (

0.0005\text{ atm/m}

) by the hourly descent distance (

9000\text{ m}

) to obtain

0.0005 \times 9000 = 4.5\text{ atm/hr}

.

Anahtar Kavram

Interpreting Linear Relationships in Context

Alternatif Yöntem

Write the cabin pressure as a function of time

t

, in seconds:

P(t) = 1.0 + 0.0005(2.5t) = 1.0 + 0.00125t

. The slope of this line,

0.00125\text{ atm/s}

, represents the rate of increase per second. To convert this rate to hours, multiply by

3600\text{ seconds/hour}

:

0.00125 \times 3600 = 4.5\text{ atm/hr}

.

Tahmini Süre:2m 0s

Soru 233Soru →

In the $xy$ -plane, the graph of the linear equation $3x + 4y = k$ , where $k$ is a positive constant, is tangent to the circle with equation $x^2 + y^2 - 2x - 4y = 20$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The value of the constant

k

is

36

.

To find the value of

k

where the line

3x + 4y = k

is tangent to the circle, we first rewrite the circle's equation

x^2 + y^2 - 2x - 4y = 20

in standard form by completing the square:

(x^2 - 2x + 1) + (y^2 - 4y + 4) = 20 + 1 + 4

, which simplifies to

(x - 1)^2 + (y - 2)^2 = 25

. This is a circle centered at

(1, 2)

with a radius of

5

. A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. Using the distance formula

\frac{|Ax_0 + By_0 - C|}{\sqrt{A^2 + B^2}}

for the point

(1, 2)

and the line

3x + 4y - k = 0

, we set up the equation:

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

. This simplifies to

\frac{|11 - k|}{5} = 5

, or

|11 - k| = 25

. Solving the absolute value equation gives

11 - k = 25 \Rightarrow k = -14

and

11 - k = -25 \Rightarrow k = 36

. Since the problem specifies that

k

is a positive constant, the correct answer is

36

.

Adım Adım Çözüm

1

Rewrite the circle's equation by completing the square for both variables.

(x - 1)^2 + (y - 2)^2 = 25

To identify the center and radius of the circle.

2

Identify the center and radius from the standard form of the circle's equation.

Center is

(1, 2)

and radius is

5

.

The standard form of a circle is

(x-h)^2 + (y-k)^2 = r^2

.

3

Apply the condition for tangency: the perpendicular distance from the center

(1, 2)

to the line

3x + 4y - k = 0

must equal the radius

5

.

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

A line is tangent to a circle if and only if the distance from the center to the line equals the radius.

4

Simplify the distance equation and solve the resulting absolute value equation for

k

.

|11 - k| = 25

, yielding

k = -14

or

k = 36

.

To find all mathematically possible values of the constant

k

.

5

Select the positive value for

k

as specified by the problem constraints.

k = 36

The problem states that

k

is a positive constant.

Anahtar Kavram

The relationship between a line and a circle in a nonlinear system, specifically using the distance from the center to a tangent line to solve for an unknown constant.

Soru 234Soru →

During a chemistry experiment, the temperature of a liquid increases at a constant rate. The temperature of the liquid is $24^\circ\text{C}$ at $5$ minutes after the experiment starts, and $48^\circ\text{C}$ at $15$ minutes after the experiment starts. The temperature $T$ , in degrees Celsius, of the liquid $t$ minutes after the experiment starts can be modeled by the equation $T = mt + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer is 12. By translating the given information into two points on the line,

(5, 24)

and

(15, 48)

, we can find the slope

m = \frac{48 - 24}{15 - 5} = 2.4

. Substituting this slope and the point

(5, 24)

into

T = mt + b

yields

24 = 2.4(5) + b

, which simplifies to

24 = 12 + b

. Solving for

b

gives

b = 12

.

Adım Adım Çözüm

1

Identify the data points representing the relationship between time and temperature.

The two data points are

(t_1, T_1) = (5, 24)

and

(t_2, T_2) = (15, 48)

.

These points represent coordinates

(t, T)

on the line representing the temperature over time.

2

Calculate the slope

m

of the linear equation.

m = \frac{48 - 24}{15 - 5} = \frac{24}{10} = 2.4

The slope represents the constant rate of temperature increase per minute.

3

Substitute the slope

m

and the coordinates of one point into the equation

T = mt + b

to solve for the y-intercept

b

.

24 = 2.4(5) + b \implies 24 = 12 + b \implies b = 12

This determines the value of the constant

b

, which corresponds to the initial temperature of the liquid.

Anahtar Kavram

Determining the y-intercept of a linear model in two variables given two points.

Soru 235Soru →

If $2^{2x+2} - 2^{2x} = 12$ for some real number $x$ , what is the value of $2^{4x}$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

By applying the product rule of exponents, the expression

2^{2x+2}

can be rewritten as

2^{2x} \cdot 2^2

, or

4 \cdot 2^{2x}

. Substituting this into the given equation yields

4 \cdot 2^{2x} - 2^{2x} = 12

. Factoring out

2^{2x}

gives

2^{2x}(4 - 1) = 12

, which simplifies to

3 \cdot 2^{2x} = 12

. Dividing both sides of the equation by 3 results in

2^{2x} = 4

. Since

2^{4x}

can be written as

(2^{2x})^2

, substituting 4 for

2^{2x}

gives

4^2 = 16

. Alternatively, solving

2^{2x} = 4

gives

2x = 2

, which means

x = 1

. Substituting

x = 1

into

2^{4x}

yields

2^{4(1)} = 2^4 = 16

.

Adım Adım Çözüm

1

Rewrite

2^{2x+2}

using exponent properties.

2^{2x+2} = 2^{2x} \cdot 2^2 = 4 \cdot 2^{2x}

To express both exponential terms with the same base power, allowing them to be factored or combined.

2

Substitute this back into the equation and factor out the common term

2^{2x}

.

2^{2x}(4 - 1) = 12 \implies 3 \cdot 2^{2x} = 12

To isolate the exponential expression.

3

Solve for

2^{2x}

by dividing both sides of the equation by 3.

2^{2x} = 4

To find the value of the exponential term.

4

Express

2^{4x}

in terms of

2^{2x}

and evaluate.

2^{4x} = (2^{2x})^2 = 4^2 = 16

To find the final requested value using the exponent rule

(a^m)^n = a^{mn}

.

Anahtar Kavram

Factoring exponential equations and applying power of a power exponent rules

Soru 236Soru →

A projectile is launched from a platform. Its height $h(t)$ , in meters, $t$ seconds after launch is modeled by the function $h(t) = -5t^2 + bt + 12$ , where $b$ is a positive constant. If the projectile reaches a maximum height of $32$ meters, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The correct answer is 20.

To find the maximum height of the projectile, we locate the vertex of the quadratic function

h(t) = -5t^2 + bt + 12

. The

t

-coordinate of the vertex is given by

t = -\frac{B}{2A} = -\frac{b}{2(-5)} = \frac{b}{10}

. Substituting

t = \frac{b}{10}

into

h(t)

and setting the height to

32

meters gives

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

. Simplifying this equation results in

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which reduces to

\frac{b^2}{20} = 20

. Solving for

b

gives

b^2 = 400

. Because

b

must be a positive constant,

b = 20

.

Adım Adım Çözüm

1

Identify the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{B}{2A}

for the quadratic function

h(t) = At^2 + Bt + C

.

t = \frac{b}{10}

The vertex of a downward-opening parabola represents its maximum value. For

h(t) = -5t^2 + bt + 12

, the coefficients are

A = -5

and

B = b

.

2

Substitute the time

t = \frac{b}{10}

back into the height function

h(t)

and set the expression equal to the maximum height of

32

meters.

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

At the maximum height, the height of the projectile is

32

meters, which corresponds to the value of the function at the vertex time.

3

Simplify the equation to solve for

b^2

.

\frac{b^2}{20} = 20

Squaring the fraction yields

-5\left(\frac{b^2}{100}\right) + \frac{b^2}{10} + 12 = 32

. Simplifying the coefficients leads to

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which simplifies to

\frac{b^2}{20} = 20

.

4

Solve the equation for

b

.

b = 20

Multiplying both sides by

20

gives

b^2 = 400

. Taking the square root of both sides gives

b = \pm 20

. Since

b

is specified to be a positive constant,

b = 20

.

Anahtar Kavram

Finding the vertex of a quadratic function to determine maximum values in a real-world context.

Tahmini Süre:2m 0s

Soru 237Soru →

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

,

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

,

2

, and

3

), we see that the maximum possible value of

x

is

3

.

Adım Adım Çözüm

1

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

.

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

2

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

3

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

.

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

4

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

,

(0, 5)

,

(2, 0)

, and

(3, 2)

.

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 238Soru →

\begin{aligned} x - y &= 3 \\ y &= x^2 - 7x + 12 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $xy$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

By substituting the linear equation

y = x - 3

into the quadratic equation, we obtain a single quadratic equation

x^2 - 8x + 15 = 0

. Factoring this equation yields the solutions

x = 3

and

x = 5

. The corresponding

y

-coordinates are

y = 0

and

y = 2

, respectively. Since the problem specifies that

y > 0

, we choose the solution

(5, 2)

. The product of the coordinates is

5 \cdot 2 = 10

.

Adım Adım Çözüm

1

Express

y

in terms of

x

from the linear equation.

y = x - 3

This allows for substitution into the second equation.

2

Substitute

y = x - 3

into the quadratic equation.

x - 3 = x^2 - 7x + 12

To create a single quadratic equation in terms of

x

.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x + 15 = 0

Subtracting

x

and adding

3

to both sides enables factoring.

4

Factor the quadratic expression.

(x - 3)(x - 5) = 0

To find the roots of the equation, which are

x = 3

and

x = 5

.

5

Find the corresponding

y

-values for each

x

-value.

For

x = 3

,

y = 3 - 3 = 0

. For

x = 5

,

y = 5 - 3 = 2

.

To obtain the complete coordinate pairs of the intersection points.

6

Apply the constraint

y > 0

to select the correct solution pair.

The solution

(5, 2)

is selected since its

y

-coordinate is greater than

0

.

The other solution,

(3, 0)

, has

y = 0

, which does not satisfy the constraint

y > 0

.

7

Calculate the value of

xy

for the chosen solution.

xy = 5 \cdot 2 = 10

To find the final requested value.

Anahtar Kavram

Solving a nonlinear system of equations containing a linear equation and a quadratic equation by substitution.

Tahmini Süre:1m 30s

Soru 239Soru →

If $4^{x - 1} = 8$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

2.5 (or 5/2)

The correct answer is 2.5 (or 5/2). By expressing 4 as

2^2

and 8 as

2^3

, the equation is rewritten as

(2^2)^{x-1} = 2^3

. Applying the exponent power rule simplifies this to

2^{2x-2} = 2^3

. Equating the exponents yields the linear equation

2x-2 = 3

. Solving for

x

gives

2x = 5

, which results in

x = 2.5

or

5/2

.

Adım Adım Çözüm

1

Rewrite both sides of the equation with a common base of 2.

(2^2)^{x-1} = 2^3

Expressing both bases as powers of 2 allows us to equate the exponents later.

2

Apply the power of a power exponent rule

(a^m)^n = a^{mn}

to the left side.

2^{2x-2} = 2^3

Multiplying the exponent 2 by the exponent

(x-1)

simplifies the expression to

2(x-1) = 2x-2

.

3

Equate the exponents and solve the linear equation for

x

.

x = 2.5

Since the bases are identical, their exponents must be equal, giving

2x-2 = 3

, which simplifies to

2x = 5

.

Anahtar Kavram

Solving exponential equations by finding a common base.

Soru 240Soru →

The function $f$ is defined by $f(x) = x^2 - 10x + 29$ . For what value of $x$ does $f(x)$ reach its minimum value?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The function reaches its minimum value at

x = 5

.

A quadratic function defined by

f(x) = ax^2 + bx + c

where

a > 0

reaches its minimum value at its vertex. The

x

-coordinate of the vertex is calculated using the formula

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

. For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

. Substituting these values into the formula gives

x = -\frac{-10}{2(1)} = 5

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic function in standard form

f(x) = ax^2 + bx + c

.

For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

.

To use the vertex formula

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

.

2

Apply the vertex formula to calculate the

x

-coordinate where the function reaches its minimum.

x = 5

.

Since the coefficient of

x^2

is positive (

a = 1

), the parabola opens upward, and its minimum value occurs at the vertex.

Anahtar Kavram

Finding the x-coordinate of the vertex of a quadratic function

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

Alternatif Yöntem

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