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ACT Math Unit 3: Modeling

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Unit 3: Modeling

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Question 1

A company's profit $P$ (in thousands of dollars) is modeled by $P = - 2 t^{2} + 12 t - 10$ , where $t$ is years after founding. In which year does the company reach maximum profit?

A.Year 3B.Year 6C.Year 2D.Year 5

Question 2

A researcher models the spread of a rumor in a school of 800 students with the equation $N (t) = \frac{800}{1 + 799 e ^{- 0.5 t}}$ , where $t$ is hours. Approximately how many students know the rumor after 10 hours?

(Use $e^{- 5} \approx 0.0067$ )

5

studentsB.

126

studentsC.

674

studentsD.

800

students

Question 3

A runner completes the first half of a race at 8 mph and the second half at 12 mph. What is the runner's average speed for the entire race?

(Note: the race is split by distance, not time.)

10

mphB.

9.6

mphC.

9

mphD.

11

mph

Question 4

A ball is dropped from a height of 200 feet. Each time it bounces, it reaches 60% of its previous height. What is the height of the ball after its 3rd bounce?

43.2

feetB.

72

feetC.

120

feetD.

48

feet

Question 5

A population of bacteria doubles every 3 hours. Starting with 500 bacteria, approximately how many bacteria are there after 12 hours?

2, 000

6, 000

8, 000

16, 000

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Question 6

A taxi charges a flat fee of $\$ 2.50 $pl u s$ $1.75$ per mile. Which equation gives the total cost $C$ for a ride of $m$ miles?

C = 2.50 m + 1.75

C = 1.75 m + 2.50

C = (2.50 + 1.75) m

C = 2.50 \times 1.75 m

Question 7

A phone plan costs $\$ 30 $p er m o n t h pl u s$ $0.10$ per text message. Last month, Jake paid $\$ 47.50$. How many text messages did Jake send?

170

175

180

77.5

Question 8

A farmer wants to fence a rectangular area along a river. He has 120 meters of fence and does not need to fence the side along the river. What is the maximum area he can enclose?

900

m²B.

1, 600

m²C.

1, 800

m²D.

3, 600

m²

Question 9

The height $h$ (in feet) of a projectile launched upward is given by $h (t) = - 16 t^{2} + 80 t + 6$ , where $t$ is time in seconds. At what time does the projectile hit the ground?

t = 5

secondsB.

t \approx 5.07

secondsC.

t = 2.5

secondsD.

t \approx 4.93

seconds

Question 10

A store sells two products. Product X has a profit of $\$ 5 $p er u ni t an d P r o d u c t Y ha s a p r o f i t o f$ $8$ per unit. The store can produce at most 200 units total, and must produce at least 60 units of Product X. If the store produces $x$ units of X and $y$ units of Y, which constraint set correctly models the situation?

x + y \leq 200

;

x \geq 60

;

x, y \geq 0

x + y = 200

;

x \geq 60

;

x, y \geq 0

x + y \leq 200

;

x \leq 60

;

x, y \geq 0

5 x + 8 y \leq 200

;

x \geq 60

;

x, y \geq 0

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Question 11

A car's value depreciates by 15% each year. The car was purchased for $\$ 24{,}000 $. W hi c h e x p r ess i o n g i v es t h ec a r^{'} s v a l u e$ V $a f t er$ t$ years?

V = 24000 - 0.15 t

V = 24000 (1.15)^{t}

V = 24000 (0.85)^{t}

V = 24000 (0.15)^{t}

Question 12

A savings account earns 4% annual interest, compounded annually. If $\$ 1{,}000$ is deposited today, what is the balance after 3 years?

\

1{,}120.00$B.

\

1{,}124.86$C.

\

1{,}040.00$D.

\

1{,}120.60$

ACT Math Unit 3: Modeling — Free with Answer Explanations | Test Practice Hub