Tüm alıştırma soruları

Soru 401Soru →

A manufacturing plant uses a machine lubricant at a constant rate of $0.08$ fluid ounces per second of operation. The plant operates the machinery for $8$ hours per day, $5$ days a week. The lubricant is purchased in drums, where each drum contains $15$ gallons of lubricant. How many drums of lubricant does the plant use in a period of $5$ weeks? (Note: $1\text{ gallon} = 128\text{ fluid ounces}$ )

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Cevap: 30

Cevap

The plant uses 30 drums of lubricant in 5 weeks.

To find the number of drums, calculate the total seconds of operation: 5 weeks * 5 days/week * 8 hours/day * 3,600 seconds/hour = 720,000 seconds. Multiply this by the consumption rate of 0.08 fluid ounces/second to get 57,600 fluid ounces. Convert this to gallons by dividing by 128 fluid ounces/gallon to get 450 gallons. Finally, divide by 15 gallons/drum to find that 30 drums are used.

Adım Adım Çözüm

1

Calculate the total number of operational seconds in the 5-week period.

720{}000\text{ seconds}

Multiply 5 weeks by 5 days per week, 8 hours per day, and 3,600 seconds per hour:

5 \times 5 \times 8 \times 3{}600 = 720{}000

.

2

Calculate the total volume of lubricant used in fluid ounces.

57{}600\text{ fluid ounces}

Multiply the rate of

0.08\text{ fluid ounces per second}

by the total operational seconds:

720{}000 \times 0.08 = 57{}600

.

3

Convert the total fluid ounces to gallons.

450\text{ gallons}

Divide the total fluid ounces by the conversion factor of

128\text{ fluid ounces per gallon}

:

57{}600 / 128 = 450

.

4

Calculate the total number of drums needed.

30\text{ drums}

Divide the total gallons by the capacity of a single drum (

15\text{ gallons}

):

450 / 15 = 30

.

Anahtar Kavram

Multi-step dimensional analysis and unit conversion under compound rates

Soru 402Soru →

Two environmental cleanup projects, Project A and Project B, begin treating separate, identical bodies of water that each contain $V_0$ gallons of a certain chemical. The volume of the chemical remaining in the water treated by Project A is modeled by the exponential decay function $A(t) = V_0(k)^t$ , where $t$ is the number of days since treatment began and $k$ is a constant. The volume of the chemical remaining in the water treated by Project B is modeled by the linear decay function $B(t) = V_0 - c \cdot t$ , where $c$ is a positive constant. After $1$ day of treatment, the volume of the chemical remaining in both bodies of water is the same. After $2$ days of treatment, the volume of the chemical remaining in the water treated by Project A is exactly $\frac{16}{15}$ times the volume of the chemical remaining in the water treated by Project B. If the treatment for Project B continues at this constant rate, after how many days will the chemical in Project B's water be completely removed?

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Cevap: 5

Cevap

5

Equating both models at

t = 1

gives

\frac{c}{V_0} = 1 - k

. At

t = 2

, Project A's remaining volume is

V_0 k^2

and Project B's is

V_0(2k - 1)

. Setting

k^2 = \frac{16}{15}(2k - 1)

results in the quadratic equation

15k^2 - 32k + 16 = 0

. Factoring this yields

k = 0.8

(discarding

1.33

because the scenario represents decay). The chemical in Project B's water is completely removed when

B(t) = 0 \implies t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Adım Adım Çözüm

1

Relate parameters

k

and

c

using the condition

A(1) = B(1)

.

\frac{c}{V_0} = 1 - k

Establishing a link between the rates of linear decay and exponential decay.

2

Express the remaining volumes at

t = 2

in terms of

k

.

A(2) = V_0 k^2

and

B(2) = V_0(2k - 1)

Preparing equations to apply the ratio condition at

t = 2

using a single variable.

3

Apply the condition

A(2) = \frac{16}{15}B(2)

to solve for

k

.

k = 0.8

Setting up the quadratic equation

15k^2 - 32k + 16 = 0

, factoring it to find roots

0.8

and

1.33

, and selecting

0.8

because

k < 1

for a decay process.

4

Calculate the time

t

when

B(t) = 0

.

t = 5

Solving

V_0 - c \cdot t = 0

yields

t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Anahtar Kavram

Linear and exponential decay modeling

Soru 403Soru →

A technician is monitoring the pressure of a gas inside a container during an experiment. The pressure $P$ , in kilopascals ( $\text{kPa}$ ), and the time elapsed $t$ , in minutes, are related by a linear equation. At $t = 4$ minutes, the pressure is $112\text{ kPa}$ . At $t = 12$ minutes, the pressure is $136\text{ kPa}$ . If the pressure continues to increase at this constant rate, what will the pressure be, in $\text{kPa}$ , at $t = 15$ minutes?

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Cevap: 145

Cevap

145

The relationship between pressure

P

and time

t

is linear, which can be modeled by the equation

P = mt + b

, where

m

is the rate of change (slope) and

b

is the pressure at

t = 0

. Using the points

(4, 112)

and

(12, 136)

, the slope is calculated as

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

. Substituting the point

(4, 112)

and

m = 3

into the equation

P = mt + b

yields

112 = 3(4) + b

, which simplifies to

112 = 12 + b

, so

b = 100

. The linear equation is

P = 3t + 100

. Substituting

t = 15

into this equation gives

P = 3(15) + 100 = 45 + 100 = 145

.

Adım Adım Çözüm

1

Calculate the rate of change (slope,

m

) using the two given coordinate points

(4, 112)

and

(12, 136)

.

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

To find the constant rate at which the pressure is increasing per minute.

2

Set up the linear equation using the point-slope form

P - P_1 = m(t - t_1)

with the point

(4, 112)

.

P - 112 = 3(t - 4) \Rightarrow P = 3t + 100

To establish the linear relationship between pressure and time.

3

Substitute

t = 15

into the linear equation to find the pressure at

15

minutes.

P = 3(15) + 100 = 145

To determine the pressure at the requested time of

15

minutes.

Anahtar Kavram

Finding and applying a linear equation in two variables from two points.

Soru 404Soru →

The table below shows some values of the exponential function $f$ , where $f(t) = p \cdot q^t$ for constants $p$ and $q$ .

$t$	$f(t)$
$0$	$80$
$2$	$180$
$4$	$405$

What is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

1.5

The correct answer is 1.5. Since the value of the function at

t = 0

is

80

, the initial value coefficient is

80

. At

t = 2

, the value is

180

, which gives the equation

80 \cdot q^2 = 180

. Solving for

q^2

yields

q^2 = 2.25

, and taking the positive square root gives

q = 1.5

.

Adım Adım Çözüm

1

Set up the general exponential equation using the initial value

f(0) = p \cdot q^0 = 80 \implies p = 80

The initial value at

t = 0

directly gives the coefficient

p

because

q^0 = 1

.

2

Substitute another point from the table to solve for the base

q

f(2) = 80 \cdot q^2 = 180 \implies q^2 = 2.25

Using the point

(2, 180)

allows us to write an equation with one variable,

q

.

3

Solve for

q

by taking the square root

q = 1.5

Since the base of an exponential function must be positive, we take the positive square root of 2.25.

Anahtar Kavram

Determining the base of an exponential function from a table of values

Soru 405Soru →

A scientist is monitoring the population of a bacterial culture in a petri dish. The population of the culture grows exponentially, doubling every $3$ hours. If the population of the culture after $9$ hours is $1,600$ , what was the initial population of the culture?

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Cevap: 200

Cevap

The initial population of the culture was

200

.

Since the bacterial culture doubles every

3

hours, the population undergoes

9 \div 3 = 3

doubling periods over a span of

9

hours. An initial population

P_0

that doubles

3

times will grow to

P_0 \times 2^3 = 8P_0

. Setting this expression equal to the final population of

1,600

gives

8P_0 = 1,600

. Dividing both sides of this equation by

8

yields

P_0 = 200

. Therefore, the initial population of the culture was

200

.

Adım Adım Çözüm

1

Determine the number of doubling periods that occurred in

9

hours.

3

doubling periods

Since the population doubles every

3

hours, dividing the total time of

9

hours by the doubling time of

3

hours gives

9 \div 3 = 3

periods.

2

Set up an equation representing the exponential growth.

P_0 \times 2^3 = 1,600

An initial population

P_0

doubling

3

times grows by a factor of

2^3

, which is equal to

8

times the initial amount.

3

Solve the equation

8P_0 = 1,600

for the initial population

P_0

.

P_0 = 200

Dividing both sides of the equation by

8

isolates

P_0

and gives the initial value.

Anahtar Kavram

Exponential growth models and solving for the initial value.

Soru 406Soru →

A quality control analyst inspected a batch of 180 microchips from two production lines, Line A and Line B. The microchips were classified as Defective, Acceptable, or Premium. Some of the data from the inspection are shown in the table below.

Production Line	Defective	Acceptable	Premium
Line A	10	50	$x$
Line B	15	$y$	35
Total	25	55	180

If a microchip is selected at random from those classified as Defective or Premium, what is the probability that the chip was produced by Line A?

Cevabı ve açıklamayı göster

Cevap: 0.375

Cevap

The correct answer is 3/8 (or 0.375).

To find the probability that a randomly selected microchip was produced by Line A given that it is classified as Defective or Premium, we restrict the sample space to only the Defective and Premium microchips. From the table, the total number of Defective microchips is 25, and the total number of Premium microchips is 55, giving a combined group size of 25 + 55 = 80 microchips. Next, we determine how many of these 80 microchips were produced by Line A. Line A produced 10 Defective microchips and x Premium microchips. Since the total number of Premium microchips is 55 and Line B produced 35, Line A produced x = 55 - 35 = 20 Premium microchips. Thus, the number of microchips produced by Line A that are Defective or Premium is 10 + 20 = 30. The conditional probability is the number of favorable outcomes divided by the restricted total outcomes, which is 30/80 = 3/8 (or 0.375).

Adım Adım Çözüm

1

Find the value of

x

(Line A Premium chips) using the total number of Premium chips.

x = 20

Since the total number of Premium chips is 55 and Line B produced 35, Line A must have produced

55 - 35 = 20

Premium chips.

2

Calculate the total number of chips produced by Line A.

Line A Total = 80

Sum the Defective, Acceptable, and Premium chips produced by Line A:

10 + 50 + 20 = 80

.

3

Calculate the total number of chips produced by Line B.

Line B Total = 100

Subtract the total number of Line A chips from the grand total of 180 chips:

180 - 80 = 100

.

4

Find the value of

y

(Line B Acceptable chips).

y = 50

Subtract the Defective (15) and Premium (35) chips of Line B from its total (100):

100 - 15 - 35 = 50

.

5

Identify the total number of chips in the conditioning category 'Defective or Premium'.

Total Defective or Premium = 80

Sum the total number of Defective chips (25) and Premium chips (55):

25 + 55 = 80

.

6

Identify the number of chips produced by Line A that are either Defective or Premium.

Favorable chips = 30

Sum the Defective chips from Line A (10) and the Premium chips from Line A (

x = 20

):

10 + 20 = 30

.

7

Calculate the probability by dividing the favorable outcomes by the total outcomes of the conditioning category.

30/80 = 3/8 = 0.375

The probability of selecting a Line A chip from the Defective or Premium group is the ratio of favorable chips to total chips in that group.

Anahtar Kavram

Conditional Probability from Two-Way Tables

Tahmini Süre:2m 30s

Soru 407Soru →

A commercial printing press has a reservoir of yellow ink. The volume of yellow ink in the reservoir, $V$ , in milliliters, after printing $p$ pages of a color brochure is modeled by the equation $V = 1,200 - 0.15p$ . According to the model, what is the decrease, in milliliters, in the volume of yellow ink in the reservoir for every 100 pages printed?

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Cevap: 15

Cevap

The volume of yellow ink in the reservoir decreases by 15 milliliters for every 100 pages printed.

In the equation

V = 1,200 - 0.15p

, the coefficient of

p

is

-0.15

. This represents the rate of change of the volume of yellow ink in the reservoir with respect to the number of pages printed. Specifically, it means the volume decreases by

0.15

milliliters for each additional page printed. To find the decrease in volume for every 100 pages printed, multiply the rate per page by 100:

0.15 \times 100 = 15

milliliters.

Adım Adım Çözüm

1

Identify the rate of change per page from the linear equation.

The rate of change is 0.15 milliliters per page.

In the linear equation

V = 1,200 - 0.15p

, the coefficient of the independent variable

p

represents the change in the dependent variable

V

for each unit increase in

p

.

2

Calculate the decrease in ink volume for 100 pages.

15 milliliters

Since the volume decreases by 0.15 milliliters for each page printed, printing 100 pages results in a total decrease of

0.15 \times 100 = 15

milliliters.

Anahtar Kavram

Interpreting the slope (rate of change) of a linear equation in context.

Soru 408Soru →

A parabola in the $xy$ -plane has vertex $(3, 18)$ and passes through the origin. If the equation of the parabola is written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

The vertex form of a quadratic function with vertex

(h, k)

is

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

. Substituting the given vertex

(3, 18)

yields the equation

y = a(x - 3)^2 + 18

. Since the parabola passes through the origin, we can substitute the point

(0, 0)

into the equation to find the value of

a

:

0 = a(0 - 3)^2 + 18 \implies 9a = -18 \implies a = -2

. Substituting

a = -2

back into the vertex form and expanding gives

y = -2(x - 3)^2 + 18 = -2(x^2 - 6x + 9) + 18 = -2x^2 + 12x

. Comparing this to the standard form

y = ax^2 + bx + c

, we identify

a = -2

and

b = 12

. The sum of these constants is

a + b = -2 + 12 = 10

.

Adım Adım Çözüm

1

Write the equation of the parabola in vertex form.

y = a(x - 3)^2 + 18

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the origin

(0, 0)

to solve for

a

.

a = -2

The parabola passes through the origin

(0, 0)

, so these coordinates must satisfy the equation.

3

Expand the vertex form equation into standard form

y = ax^2 + bx + c

.

y = -2x^2 + 12x

Expanding the equation allows us to identify the coefficients

a

and

b

directly.

4

Identify the values of

a

and

b

, and calculate

a + b

.

a + b = 10

Comparing

y = -2x^2 + 12x

to

y = ax^2 + bx + c

gives

a = -2

and

b = 12

, so

a + b = -2 + 12 = 10

.

Anahtar Kavram

Vertex form of quadratic functions and conversion to standard form

Soru 409Soru →

If $-3(2x - 5) + 4x \geq -7$ , what is the maximum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The maximum possible value of

x

is

11

.

By applying the distributive property, combining like terms, and dividing by

-2

(while reversing the inequality sign), we find that the solution is

x \leq 11

. Thus, the maximum possible value of

x

is

11

.

Adım Adım Çözüm

1

Apply the distributive property to simplify the left side of the inequality.

-6x + 15 + 4x \geq -7

Multiplying

-3

by each term inside the parentheses

(2x - 5)

yields

-6x

and

+15

.

2

Combine the variable terms on the left side.

-2x + 15 \geq -7

Combining

-6x

and

4x

gives

-2x

.

3

Subtract

15

from both sides of the inequality.

-2x \geq -22

To isolate the variable term

-2x

on the left side.

4

Divide both sides by

-2

and flip the inequality sign.

x \leq 11

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

5

Determine the maximum value from the solution set.

11

The solution set consists of all values less than or equal to

11

, so the greatest value is

11

.

Anahtar Kavram

Solving multi-step linear inequalities, including applying the distributive property and reversing the inequality sign when dividing by a negative number.

Soru 410Soru →

The function $f$ is defined by $f(x) = 3^x - 4$ . In the $xy$ -plane, the graph of the function $g$ is obtained by first reflecting the graph of $f$ across the $x$ -axis, then translating the graph vertically up by 10 units, and finally translating the graph horizontally to the right by 2 units. If $g(c) = 5$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of c is 4.

Reflecting the function

f(x) = 3^x - 4

across the

x

-axis changes its sign to

-f(x) = -3^x + 4

. Translating this graph vertically up by 10 units adds 10 to the function, yielding

-3^x + 14

. Finally, translating horizontally to the right by 2 units replaces

x

with

x - 2

, producing the function

g(x) = -3^{x-2} + 14

. Setting

g(c) = 5

gives the equation

-3^{c-2} + 14 = 5

. Subtracting 14 from both sides results in

-3^{c-2} = -9

, which simplifies to

3^{c-2} = 9

. Since

9 = 3^2

, the exponent

c-2

must equal 2, which gives

c = 4

.

Adım Adım Çözüm

1

Reflect the function

f(x) = 3^x - 4

across the

x

-axis

-f(x) = -(3^x - 4) = -3^x + 4

A reflection across the

x

-axis replaces

y

with

-y

, meaning the entire function is multiplied by

-1

.

2

Translate the reflected function vertically up by 10 units

-3^x + 4 + 10 = -3^x + 14

A vertical translation upward by

k

units adds

k

directly to the function expression.

3

Translate the function horizontally to the right by 2 units

g(x) = -3^{x-2} + 14

A horizontal translation to the right by

h

units replaces

x

with

x-h

in the function expression.

4

Set

g(c) = 5

and solve the exponential equation for

c

-3^{c-2} + 14 = 5 \implies 3^{c-2} = 9 \implies c - 2 = 2 \implies c = 4

Substitute

c

into

g(x)

, set the output to 5, isolate the exponential term, and equate exponents to find

c

.

Anahtar Kavram

Applying sequential function transformations (reflections, vertical translations, horizontal translations) algebraically and solving exponential equations.

Soru 411Soru →

In the equation $x(x - 8) = k$ , $k$ is a constant. If the product of the two real solutions to the equation is $-20$ , what is the value of the larger solution?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The larger solution to the equation is 10.

To find the larger solution, the equation is first rewritten in standard form as

x^2 - 8x - k = 0

. The product of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is

\frac{c}{a}

. Here,

a = 1

and

c = -k

, so the product of the roots is

-k

. Given that the product of the roots is

-20

, we can set up the equation

-k = -20

, which gives

k = 20

. Substituting

k = 20

back into the equation yields

x^2 - 8x - 20 = 0

. Factoring the quadratic expression gives

(x - 10)(x + 2) = 0

, which has the solutions

x = 10

and

x = -2

. The larger of these two solutions is 10.

Adım Adım Çözüm

1

Rewrite the given equation

x(x - 8) = k

in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x - k = 0

To apply quadratic properties, the equation must be in standard form.

2

Use the product of roots formula to find the value of the constant

k

.

-k = -20

, which simplifies to

k = 20

The product of the roots of a quadratic equation in standard form is the constant term divided by the leading coefficient.

3

Substitute the value of

k

back into the quadratic equation and factor it to find the two solutions.

(x - 10)(x + 2) = 0

, so

x = 10

or

x = -2

Factoring the quadratic equation allows us to find the individual roots.

4

Compare the two solutions to identify the larger value.

10

Comparing 10 and -2, 10 is the greater value.

Anahtar Kavram

Using the relationship between coefficients and the product of roots to solve a quadratic equation.

Soru 412Soru →

The table below shows the distribution of the number of public library cards owned by a group of families in a certain neighborhood.

Number of library cards	Number of families
1	4
2	5
3	2
4	1

What is the median number of public library cards owned by these families?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The median number of public library cards is 2.

To find the median, first determine the total number of families by summing the frequencies:

4 + 5 + 2 + 1 = 12

. Since the number of data points is even, the median is the average of the two middle values (

6^{\text{th}}

and

7^{\text{th}}

values). Sorting the data, the first 4 values are 1, and the next 5 values (positions 5 through 9) are 2. Since both the

6^{\text{th}}

and

7^{\text{th}}

values are 2, the median is 2.

Adım Adım Çözüm

1

Find the total number of families by adding the frequencies.

4 + 5 + 2 + 1 = 12

To determine the position of the median in the dataset, we must first find the total number of data points.

2

Identify the positions of the middle values.

The

6^{\text{th}}

and

7^{\text{th}}

values.

For an even number of data points (

n = 12

), the median is the average of the values at positions

\frac{n}{2}

and

\frac{n}{2} + 1

.

3

Find the values at the

6^{\text{th}}

and

7^{\text{th}}

positions using the frequency table.

Both the

6^{\text{th}}

and

7^{\text{th}}

values are 2.

Accumulating the frequencies from top to bottom: the first 4 values are 1, and the next 5 values (positions 5 to 9) are 2.

4

Calculate the average of the two middle values.

\frac{2 + 2}{2} = 2

The median of an even-sized dataset is the arithmetic mean of its two middle values.

Anahtar Kavram

Calculating the median from a frequency table

Soru 413Soru →

Two runners, Estela and Desmond, start at the same point on a circular track and run in opposite directions at constant speeds. The ratio of Estela's speed to Desmond's speed is $5$ to $4$ . If they start at the same time, they meet for the first time after $40$ seconds. If Desmond increases his speed by $25\%$ immediately after their first meeting, how many seconds after their first meeting will they meet for the second time?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

To find the time between the first and second meetings, we determine the track length in terms of a constant

k

. Given the speed ratio of

5

to

4

, the speeds are

5k

and

4k

. The relative speed in opposite directions is the sum of these speeds, or

9k

. In

40

seconds, they cover the track length

L = 9k \times 40 = 360k

. When Desmond increases his speed by

25\%

, his new speed is

4k \times 1.25 = 5k

, making the new relative speed

5k + 5k = 10k

. The time to meet again is the track length divided by the new relative speed:

360k / 10k = 36

seconds.

Adım Adım Çözüm

1

Represent the speeds of both runners using the given ratio of

5

to

4

.

Estela's speed is

5k

and Desmond's speed is

4k

, where

k

is a positive constant.

Using a common variable allows calculation of relative rates without needing their absolute speeds.

2

Calculate the initial relative speed of approach.

Relative speed =

5k + 4k = 9k

.

Since they run in opposite directions around the circular track, their rates of travel add up to determine their rate of approach.

3

Determine the track circumference

L

in terms of

k

.

Track length

L = 9k \times 40 = 360k

.

They meet for the first time when their combined distance traveled is exactly equal to one full lap of the track.

4

Calculate Desmond's updated speed after his speed increase.

Desmond's new speed =

4k \times (1 + 0.25) = 5k

.

His speed increases by

25\%

, which corresponds to multiplying his initial speed of

4k

by

1.25

.

5

Find the new relative speed of approach after the first meeting.

New relative speed =

5k + 5k = 10k

.

Estela's speed remains

5k

and Desmond's speed is now

5k

, so their sum gives the new combined rate.

6

Solve for the time elapsed between the first and second meetings.

Time

t = \frac{360k}{10k} = 36

seconds.

To meet a second time, they must cover the track length of

360k

again from their first meeting point at the new relative speed.

Anahtar Kavram

Solving multi-step rate-time-distance problems on circular tracks using ratios and percentage change.

Soru 414Soru →

An organic farm harvested a certain amount of wheat in 2024. In 2025, due to favorable weather, the amount of wheat harvested was $25\%$ greater than in 2024. In 2026, due to a dry season, the harvest was $12\%$ less than in 2025. If the farm harvested $880$ bags of wheat in 2026, how many bags of wheat did it harvest in 2024?

Cevabı ve açıklamayı göster

Cevap: 800

Cevap

800

The correct answer is 800. Let the 2024 harvest be

w

bags. A

25\%

increase in 2025 means the harvest became

1.25w

. A subsequent

12\%

decrease in 2026 means the harvest became

0.88

of the 2025 value, which is

1.25w \times 0.88 = 1.1w

. Given that the 2026 harvest was

880

bags, we solve

1.1w = 880

to find

w = 800

.

Adım Adım Çözüm

1

Define the variable for the 2024 harvest and write an expression for the 2025 harvest.

The harvest in 2025 is

1.25w

, where

w

is the harvest in 2024.

An increase of 25% is equivalent to multiplying the original value by

1 + 0.25 = 1.25

.

2

Write an expression for the 2026 harvest in terms of the 2024 harvest variable.

The harvest in 2026 is

1.1w

.

A decrease of 12% is equivalent to multiplying the 2025 value by

1 - 0.12 = 0.88

. Multiplying the factors gives

1.25 \times 0.88 = 1.1

.

3

Set up an equation using the 2026 harvest total of 880 and solve for the 2024 harvest variable.

w = 800

Dividing the 2026 harvest value of 880 by the cumulative rate of 1.1 yields the initial 2024 value.

Anahtar Kavram

Solving for initial values in multi-step percent change scenarios.

Alternatif Yöntem

Work backward from the final year. The 2026 harvest of

880

bags is

12\%

less than the 2025 harvest, meaning it represents

88\%

of the 2025 harvest. Thus, the 2025 harvest was

880 / 0.88 = 1000

bags. The 2025 harvest of

1000

bags was

25\%

greater than the 2024 harvest, meaning it represents

125\%

of the 2024 harvest. Thus, the 2024 harvest was

1000 / 1.25 = 800

bags.

Tahmini Süre:1m 30s

Soru 415Soru →

The volume of a right circular cylinder is $72\pi$ . If the radius of the base of the cylinder is $3$ , what is the height of the cylinder?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The height of the cylinder is 8.

The volume

V

of a right circular cylinder is given by the formula

V = \pi r^2 h

, where

r

is the base radius and

h

is the height. By substituting the given volume of

72\pi

and radius of

3

, we obtain the equation

72\pi = \pi (3)^2 h

. Simplifying the right side gives

72\pi = 9\pi h

. Dividing both sides of the equation by

9\pi

isolates the height, giving

h = 8

.

Adım Adım Çözüm

1

Use the formula for the volume of a right circular cylinder.

V = \pi r^2 h

This formula relates the volume, radius, and height of a cylinder.

2

Substitute the known values into the volume formula.

72\pi = \pi (3)^2 h

To form an equation with the unknown height.

3

Simplify the squared radius and isolate the variable.

h = 8

Simplifying

3^2

gives

9

. Dividing

72\pi

by

9\pi

isolates the height.

Anahtar Kavram

Solving for an unknown dimension using the volume formula of a cylinder.

Tahmini Süre:45s

Soru 416Soru →

A quality control manager at a manufacturing plant selected a random sample of $150$ lightbulbs from a batch of $3,000$ lightbulbs produced on a certain day. The manager found that $6$ of the selected lightbulbs were defective. Based on this sample, what is the estimated number of defective lightbulbs in the entire batch of $3,000$ ?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The correct answer is

120

. Since the sample of lightbulbs was selected at random, the proportion of defective lightbulbs in the sample can be used to estimate the proportion of defective lightbulbs in the entire batch. The proportion of defective lightbulbs in the sample is

\frac{6}{150} = 0.04

. Multiplying this proportion by the total number of lightbulbs in the batch gives the estimated number of defective lightbulbs:

0.04 \times 3,000 = 120

.

Adım Adım Çözüm

1

Calculate the proportion of defective lightbulbs in the sample.

The sample proportion is

\frac{6}{150} = 0.04

.

This establishes the rate of defects in the sample to be generalized to the entire batch.

2

Estimate the total number of defective lightbulbs in the batch.

The estimate is

0.04 \times 3,000 = 120

.

Scaling the sample proportion to the population size yields the expected number of defective lightbulbs in the entire batch.

Anahtar Kavram

Generalizing results from a random sample to estimate a population parameter

Soru 417Soru →

A vertical farm uses an automated nutrient delivery system for a crop of lettuce. The total volume of nutrient solution, $V$ , in liters, remaining in the system's reservoir is modeled as a linear function of the time $t$ , in hours, since the system started running for the day. The table below shows several values of $t$ and the corresponding values of $V$ .

Time, $t$ (hours)	Volume, $V$ (liters)
3	415
5	385
8	340
12	280

According to the model, what was the initial volume of nutrient solution, in liters, in the reservoir when the system started running?

Cevabı ve açıklamayı göster

Cevap: 460

Cevap

460

The correct answer is 460. The remaining volume of nutrient solution,

V

, is a linear function of time,

t

, which can be written in the form

V = mt + b

, where

m

is the rate of change (slope) and

b

is the initial volume (y-intercept). The rate of change can be found using any two points from the table, for example,

(3, 415)

and

(5, 385)

:

m = \frac{385 - 415}{5 - 3} = \frac{-30}{2} = -15

liters per hour. Using the point

(3, 415)

and substituting

m = -15

,

t = 3

, and

V = 415

into the equation

V = mt + b

yields

415 = -15(3) + b

. Simplifying this expression gives

415 = -45 + b

, and adding 45 to both sides gives

b = 460

. Therefore, the initial volume of nutrient solution in the reservoir was 460 liters.

Adım Adım Çözüm

1

Calculate the constant rate of consumption (slope) of the nutrient solution.

The rate of consumption is 15 liters per hour.

The relationship between volume and time is linear. The slope

m

can be calculated from two coordinates from the table,

(3, 415)

and

(5, 385)

, as

m = \frac{385 - 415}{5 - 3} = -15

liters per hour.

2

Determine the initial volume of nutrient solution (the y-intercept) using the rate and a data point.

The initial volume is 460 liters.

Substitute the slope

m = -15

, the time

t = 3

, and the remaining volume

V = 415

into the slope-intercept equation

V = mt + b

. Solving

415 = -15(3) + b

gives

b = 460

.

Anahtar Kavram

Interpreting the y-intercept of a linear function in context as the initial value of the dependent variable when the independent variable is 0.

Tahmini Süre:1m 30s

Soru 418Soru →

A certain antique watch increases in value by a constant percent each year. The watch was purchased for $150$ , and its value after $2$ years is $216$ . What is the annual percent increase in the value of the watch?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The annual percent increase in the value of the watch is 20.

Because the watch increases in value by a constant percent each year, its value can be modeled by the exponential growth equation

V(t) = P(1+r)^t

, where

P

is the initial purchase price,

r

is the annual growth rate as a decimal, and

t

is the number of years. Substituting the given values

P = 150

,

t = 2

, and

V(2) = 216

yields

150(1+r)^2 = 216

. Dividing both sides by

150

gives

(1+r)^2 = 1.44

. Taking the positive square root of both sides results in

1+r = 1.2

, which simplifies to

r = 0.2

. Converting

0.2

to a percentage gives an annual percent increase of

20\%

. Therefore, the correct numeric answer is

20

.

Adım Adım Çözüm

1

Set up the exponential growth equation using the given values.

150(1+r)^2 = 216

Since the value increases by a constant percent each year, the growth is exponential and modeled by

V(t) = P(1+r)^t

.

2

Divide both sides of the equation by

150

to isolate the exponential base term.

(1+r)^2 = 1.44

Dividing

216

by

150

simplifies the equation to solve for the growth factor.

3

Take the positive square root of both sides of the equation.

1 + r = 1.2

Taking the square root of

1.44

isolates the expression

1+r

.

4

Solve for

r

and convert the decimal to a percentage.

r = 0.2

, which is

20\%

Subtracting

1

from both sides gives the annual rate

r = 0.2

, and multiplying by

100

gives the percent increase.

Anahtar Kavram

Solving for the growth rate in an exponential growth model

Alternatif Yöntem

Alternatively, since the question asks for a simple percent increase, you can test a few straightforward percentages. For example, if the annual increase were

10\%

, the value after 1 year would be

150 \times 1.10 = 165

, and after 2 years would be

165 \times 1.10 = 181.5

, which is too low. Testing

20\%

, the value after 1 year is

150 \times 1.20 = 180

, and after 2 years is

180 \times 1.20 = 216

. This matches the given value exactly.

Tahmini Süre:45s

Soru 419Soru →

An industrial water filtration system filters water at a constant rate of $0.5\text{ liters per second}$ . The system consumes a chemical powder at a rate of $2.5\text{ milligrams}$ per gram of impurities removed from the water. If the water contains an average of $0.12\text{ grams}$ of impurities per liter, how many grams of the chemical powder does the system consume during $4\text{ hours}$ of continuous operation?

Cevabı ve açıklamayı göster

Cevap: 2.16

Cevap

The system consumes

2.16\text{ grams}

of chemical powder during the

4

hours of operation.

The correct answer is

2.16

. First, convert the

4

hours of continuous operation into seconds:

4 \times 3,600 = 14,400

seconds. Next, multiply this duration by the filtration rate to find the total water filtered:

14,400 \times 0.5 = 7,200

liters. Multiply the volume by the impurity concentration to find the total mass of impurities:

7,200 \times 0.12 = 864

grams. Then, find the mass of chemical powder consumed:

864 \times 2.5 = 2,160

milligrams. Finally, convert milligrams to grams by dividing by

1,000

, resulting in

2.16

grams.

Adım Adım Çözüm

1

Convert the operating time from hours to seconds.

14,400\text{ seconds}

Since the filtration rate is given in liters per second, the total operating time must be converted from hours to seconds to align units. There are

3,600

seconds in one hour.

2

Calculate the total volume of water filtered during the operation.

7,200\text{ liters}

Multiply the operating time in seconds by the filtration rate of

0.5

liters per second.

3

Calculate the total mass of impurities removed from the filtered water.

864\text{ grams}

Multiply the total volume of filtered water by the concentration rate of

0.12

grams of impurities per liter.

4

Determine the mass of chemical powder consumed in milligrams.

2,160\text{ milligrams}

Multiply the total mass of impurities in grams by the consumption rate of

2.5

milligrams of chemical powder per gram of impurities.

5

Convert the mass of chemical powder from milligrams to grams.

2.16\text{ grams}

Divide the mass in milligrams by

1,000

because there are

1,000

milligrams in one gram.

Anahtar Kavram

Dimensional analysis and multi-step unit conversion involving rates

Soru 420Soru →

The table below summarizes the results of a survey about extracurricular participation among 11th-grade and 12th-grade students at a high school.

Grade	Participates	Does not participate
11th Grade	$45$	$30$
12th Grade	$55$	$20$

Based on the table, if a student who participates in extracurricular activities is selected at random, what is the probability that the student is in the 11th grade? (Express your answer as a decimal or a fraction.)

Cevabı ve açıklamayı göster

Cevap: 0.45

Cevap

0.45 (or 9/20)

To find the probability that a student is in the 11th grade given that they participate in extracurricular activities, the sample space is restricted to only those students who participate. According to the table, the total number of students who participate is

45 + 55 = 100

. Out of these

100

students,

45

are in the 11th grade. The probability is therefore

\frac{45}{100}

, which can be written as

\frac{9}{20}

or

0.45

.

Adım Adım Çözüm

1

Determine the size of the restricted sample space by finding the total number of students who participate in extracurricular activities.

100

The probability is conditional on selecting a student who participates in extracurricular activities, so only the 'Participates' column is considered.

2

Identify the number of students within this restricted group who are in the 11th grade.

45

We count the number of students who are both in the 11th grade and participate in extracurricular activities.

3

Calculate the probability by dividing the number of favorable outcomes by the size of the restricted sample space.

45/100 = 9/20 = 0.45

The probability is the ratio of 11th-grade participants to the total participants.

Anahtar Kavram

Conditional Probability from a Two-Way Table

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Alternatif Yöntem

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Alternatif Yöntem

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