Applied-Algebra Questions and Answers

This question asks us to compare the heights of two balls at a specific time:

t=0.9

The graph shows:

Ball 1 as the solid blue curve.

Ball 2 as the dashed blue curve.

To determine which ball was lower at t=0.9, we look at the vertical positions of both curves when the time is 0.9seconds.

At approximately t=0.9:

" height of Ball 1 " < " height of Ball 2 "

This means Ball 1 is lower than Ball 2 at that time.

The correct reasoning must say both:

Ball 1 was lower, and

Ball 1’s height was less than Ball 2’s height at t=0.9.

That matches option A.

From d=0to about d=2, the graph is decreasing, but it is flattening as it approaches a minimum. This means the number of raffle tickets sold is decreasing at a slower rate:

" decreases slower and slower "

After the minimum, from about d=2to d=4.4, the graph increases and becomes steeper. This means the number of raffle tickets sold is increasing at a faster rate:

" increases faster and faster "

So the correct description is:

" The number of raffle tickets decreases slower and slower and then increases faster and faster. "

The function M(t)represents the company’s market share after tyears.

The horizontal axis represents:

t= " time in years "

The vertical axis represents:

M(t)= " market share percentage "

We need to find when the market share reaches:

40%

From the graph, locate 40on the vertical axis. Then move horizontally to the blue curve and read the corresponding value on the horizontal axis.

The curve reaches 40%at approximately:

t=7.4

So the company should expect to have a market share of 40%after about:

7.4 " years "

Therefore, the correct answer is:

▭ ( " C " )

This situation can be modeled using a linear relationship because each sack has the same weight.

We are given:

After 3sacks are removed, the total weight is 116pounds.

After 6sacks are removed, the total weight is 101pounds.

From 3 sacks removed to 6 sacks removed, the number of removed sacks increases by:

6-3=3

During that time, the total weight decreases from 116pounds to 101pounds:

116-101=15

So removing 3 additional sacks decreases the total weight by 15 pounds.

Now divide to find the weight of one sack:

15/3=5

So each sack weighs:

5 " pounds "

Check:

If 3 more sacks are removed and each sack weighs 5 pounds, the total weight should decrease by:

3(5)=15

116-15=101

This matches the given information.

We are told that the number of property sales this year is expected to be 8 times the number of property sales last year.

Let:

x= " number of properties sold last year "

and

H(x)= " number of property sales this year "

Since this year’s sales are 8 times last year’s sales, the function rule is:

H(x)=8x

The question says that last year’s property sales were:

x=360

Substitute 360into the function:

H(360)=8(360)

H(360)=2,880

So the correct notation is:

H(360)=2,880

This means: if 360 properties were sold last year, then 2,880 property sales are expected this year.

Therefore, the correct answer is:

▭ ( " C " )

The function a(x)represents the total value of an investment account.

The horizontal axis represents:

x= " time in years "

The vertical axis represents:

a(x)= " total account value in dollars "

We are asked to find the total value when:

x=4

To answer this from the graph:

Locate x=4on the horizontal axis.

Move vertically upward until reaching the blue graph.

Read the corresponding value on the vertical axis.

From the graph, when x=4, the value is slightly above $1,800, approximately:

a(4)≈1,830.59

So the total value of the investment account after 4 years is approximately:

$1,830.59

The function is:

P(t)=24t+185

This is a linear function in the form:

P(t)=mt+b

where:

m= " rate of change "

and

b= " initial value "

In the function:

P(t)=24t+185

the coefficient of tis:

24

So the rate of change is:

24 " fish per month "

Because 24is positive, the fish population is increasing.

The number 185is not the rate of change. It represents the starting fish population at the beginning of the year, when t=0:

P(0)=24(0)+185=185

Therefore, the correct interpretation is:

" The number of fish in the lake is increasing at a constant rate of 24 fish per month. "

So the correct answer is:

▭ ( " B " )

The graph shows a decreasing exponential function.

The horizontal axis represents:

" Time in months "

The vertical axis represents:

" New user accounts per day "

The graph starts near 7,000new user accounts per day and decreases over time. However, it does not decrease toward 0. Instead, the graph levels off near a horizontal value around:

y=2,500

This horizontal value is the horizontal asymptote.

A horizontal asymptote shows the long-term value the function approaches. Since the asymptote is above 0, the model predicts that the number of daily new user accounts approaches a positive value, not zero.

So the justified statement is:

" The number of daily new user accounts will not decrease to 0. "

This question asks us to interpret a value from a graph.

The horizontal axis represents:

Days

The vertical axis represents:

Practice duration in minutes

We need to find the practice duration on day 8.

To do this:

Locate 8 on the horizontal axis.

Move vertically until you reach the blue graph.

Read the corresponding value on the vertical axis.

From the graph, when:

x=8

the blue graph is at:

y=115

So the musician practiced for:

115 minutes

Since xrepresents the number of years since 2003:

2003 ⇒ x=0

2007 ⇒ x=4

The average rate of change from 2003 to 2007 is:

(f(4)-f(0))/(4-0)

Find f(0):

f(0)=280× 〖 1.08 〗 ^0

f(0)=280

Find f(4):

f(4)=280× 〖 1.08 〗 ^4

f(4)≈380.94

Now calculate:

(380.94-280)/4

=100.94/4

≈25.23

Rounded to the nearest whole number:

25

Since the value is positive, the number of subscribers is increasing.

So the correct interpretation is:

" The number of subscribers increases at an average rate of 25 subscribers per year. "

Therefore, the correct answer is:

▭ ( " C " )

From t=6.2to about t=8.5, the graph is increasing, but it is becoming less steep as it approaches a local maximum.

That means the number of customers is:

" increasing slower and slower "

After the local maximum, from about t=8.5to t=10.1, the graph begins to decrease and becomes steeper downward.

That means the number of customers is:

" decreasing faster and faster "

So the full description is:

" The number of customers increases slower and slower and then decreases faster and faster. "

Therefore, the correct answer is:

▭ ( " D " )

The table gives the percent of customers with fiber internet for each year since 2020.

Since 2023 is:

2023−2020=3

we use the row where:

Years since 2020=3

From the table, the percent of customers with fiber internet in 2023 is:

57%

The company has approximately:

1,050

customers each year.

Now find 57% of 1,050:

0.57×1,050=598.5

The closest answer choice is:

598

So approximately 598 customers had fiber internet in 2023.

Therefore, the correct answer is:

C

Since xrepresents the number of years since 2004:

2013-2004=9

so:

2013 ⇒ x=9

Also:

2018-2004=14

so:

2018 ⇒ x=14

The average yearly rate of change from 2013 to 2018 is:

(f(14)-f(9))/(14-9)

Now evaluate the function:

f(9)=330× 〖 1.04 〗 ^9

f(9)≈469.37

f(14)=330× 〖 1.04 〗 ^14

f(14)≈571.13

Now calculate the average rate of change:

(571.13-469.37)/5

=101.76/5

=20.35

So the painting’s value increased by an average of about:

$20.35 " per year "

Therefore, the correct answer is:

▭ ( " A " )

This situation can be modeled with a linear relationship because each box has the same weight.

We are given:

4 " boxes " →116 " pounds "

and

9 " boxes " →161 " pounds "

The weight increases because more boxes are added.

First, find the change in total weight:

161-116=45

Then find the change in the number of boxes:

9-4=5

Since 5 additional boxes increased the total weight by 45 pounds, the weight of each box is:

45/5=9

So each box weighs:

9 " pounds "

Check:

If each box weighs 9 pounds, then adding 5 more boxes adds:

5(9)=45

116+45=161

This matches the given information.

This question asks us to interpret a graph showing cost and revenue.

In Applied Algebra, the break-even point is where:

" Revenue " = " Cost "

On a graph, this happens where the revenue line and the cost line intersect.

From the graph:

The solid blue line represents revenue.

The dashed blue line represents cost.

The break-even point occurs where these two lines cross.

Looking carefully at the graph, the two lines intersect at approximately:

x=6

The horizontal axis represents the number of units, meaning the number of down quilts sold.

So the person needs to sell about:

6 " quilts "

to break even.

After selling more than 6 quilts, the revenue line is above the cost line, meaning the person begins making a profit.

The graph models the number of widgets the manufacturing team can produce each day.

The horizontal axis represents:

" Days "

The vertical axis represents:

" Widgets produced each day "

The two labeled points are:

(16│9.1)

and

(18│12.4)

This means:

On day 16, the team could produce about 9.1widgets per day.

On day 18, the team could produce about 12.4widgets per day.

The average rate of change is:

(12.4-9.1)/(18-16)

=3.3/2

=1.65

Rounded to the nearest tenth:

1.7

So from day 16 to day 18, the team’s production increased by an average of about:

1.7 " widgets each day "

The function is:

G(t)=1,600× 〖 1.31 〗 ^t

This is an exponential growth function because the base is greater than 1:

1.31 > 1

We need to find the population after 25 minutes, so substitute:

t=25

into the function:

G(25)=1,600× 〖 1.31 〗 ^25

Evaluate the exponent and multiply:

G(25)≈1,367,421.41

Since the population is a number of bacteria cells, round to the nearest whole number:

G(25)≈1,367,421

We are told that supervisor B has 13 more employees than supervisor A.

Let:

n= " number of employees under supervisor A "

and

F(n)= " number of employees under supervisor B "

The phrase “13 more than” means we add 13:

F(n)=n+13

So, if supervisor A has nemployees, supervisor B has 13 additional employees.

For example, if supervisor A had 20 employees, then supervisor B would have:

F(20)=20+13=33

Therefore, the correct answer is:

▭ ( " C " )

The graph shows a linear relationship between:

w= " water in cups "

and

l= " lemon juice in cups "

The line passes through the origin, so the relationship has the form:

l=mw

where mis the rate of change.

From the graph, a clear point on the line is approximately:

(8│5)

This means when there are 8 cups of water, there are 5 cups of lemon juice.

The rate of change is:

m= " change in lemon juice " / " change in water "

m=5/8

So the relationship is:

l=5/8 w

This means the amount of lemon juice must be:

5/8

of the amount of water.

The graph shows a curved, downward-opening function. This type of graph is commonly associated with a quadratic polynomial function.

The maximum value of a downward-opening parabola occurs at its highest point, called the vertex.

From the graph, the highest point occurs at approximately:

t=9.5

This means the museum reaches its maximum daily profit approximately:

9.5 " years after opening "

The vertical axis represents daily profit in hundreds of dollars. From the graph, the maximum P(t)-value is approximately:

16.5

Since the profit is measured in hundreds of dollars:

16.5×100=1650

So the maximum daily profit is approximately:

$1,650

Therefore, the correct interpretation is:

" Approximately 9.5 years after opening, a maximum daily profit of approximately " $1,650 " was earned. "

The contractor charges a flat rate plus a fixed rate per hour, so this situation is modeled by a linear function:

C(t)=mt+b

where:

m= " hourly rate "

and

b= " flat rate "

We are given two points:

(3│220)

and

(5│340)

These mean:

A 3-hour job costs $220.

A 5-hour job costs $340.

First, find the hourly rate using slope:

m=(340-220)/(5-3)

m=120/2

m=60

So the contractor charges:

$60 " per hour "

Now the function has the form:

C(t)=60t+b

Use the point (3│220)to find the flat rate:

220=60(3)+b

220=180+b

b=40

So the function is:

C(t)=60t+40

Check with t=5:

C(5)=60(5)+40

C(5)=300+40

C(5)=340

This matches the given information.

The function c(t)gives the number of active shopping carts on a website.

Because c(t)represents active carts, the difference between c(6)and c(8)will be measured in:

" active carts "

not hours.

First evaluate c(6):

c(6)=-0.01(6)^4+0.25(6)^3-2.33(6)^2+9.02(6)+8

c(6)=19.28

Now evaluate c(8):

c(8)=-0.01(8)^4+0.25(8)^3-2.33(8)^2+9.02(8)+8

c(8)=18.08

Now find the difference:

19.28-18.08=1.20

This is approximately:

1 " active cart "

Therefore, the correct answer is:

▭ ( " B " )

This question asks for the break-even point, which is the point where:

Revenue=Cost

On the graph:

The solid blue line represents revenue.

The dashed blue line represents cost.

The revenue line starts below the cost line, meaning the person is not yet covering costs. The revenue line eventually crosses the cost line at a little more than 4 units.

Since the question asks for the minimum number of livestock feeders that must be sold, we need a whole number of feeders.

The graph shows the intersection occurs between:

4 and 5

So selling 4 feeders is not enough, but selling 5 feeders is enough for revenue to cover the costs.

Therefore, the correct answer is:

C