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Math  /  Algebra

QuestionBetween 2000 and 2020, the population of Mathville could be modeled by the function m(t)=100t3m(t)=100 \sqrt[3]{t}, where m(t)m(t) is the number of people in Mathville, and tt is the number of years since 2000. Between those same years, the population of Calcfield could be modeled by the function c(t)=18tc(t)=18 t. A. Graph each function on graph paper or a neatly made coordinate grid by hand. Be sure to consider an appropriate domain for the functions as you make your graph. B. Approximately where do the functions intersect? What does this point of intersection represent? C. Write and solve an equation to algebraically confirm where the two functions intersect. Show your work. D. Write 2-3 complete sentences comparing the relative populations of the cities over time (10 points)

Studdy Solution

STEP 1

1. The population of Mathville is modeled by m(t)=100t3 m(t) = 100 \sqrt[3]{t} .
2. The population of Calcfield is modeled by c(t)=18t c(t) = 18t .
3. t t represents the number of years since 2000, so t t ranges from 0 to 20.
4. We need to graph the functions, find the intersection, and compare populations.

STEP 2

1. Graph the functions.
2. Determine the approximate intersection point.
3. Algebraically find the intersection point.
4. Compare the populations over time.

STEP 3

Graph the function m(t)=100t3 m(t) = 100 \sqrt[3]{t} on a coordinate grid, considering the domain 0≤t≤20 0 \leq t \leq 20 .
Graph the function c(t)=18t c(t) = 18t on the same coordinate grid, considering the domain 0≤t≤20 0 \leq t \leq 20 .

STEP 4

Visually inspect the graph to determine the approximate intersection point of the two functions.

STEP 5

Set the equations equal to find the intersection point algebraically:
100t3=18t 100 \sqrt[3]{t} = 18t

STEP 6

Cube both sides to eliminate the cube root:
1003⋅t=(18t)3 100^3 \cdot t = (18t)^3
Simplify and solve for t t :
1000000t=5832t3 1000000t = 5832t^3
Divide both sides by t t (assuming t≠0 t \neq 0 ):
1000000=5832t2 1000000 = 5832t^2
Solve for t2 t^2 :
t2=10000005832 t^2 = \frac{1000000}{5832}
t=10000005832 t = \sqrt{\frac{1000000}{5832}}
Calculate t t :
t≈13.12 t \approx 13.12

STEP 7

The intersection point represents the year when the populations of Mathville and Calcfield are equal. This occurs approximately 13 years after 2000, which is around the year 2013.
Compare the populations over time:
Initially, Calcfield's population grows linearly, while Mathville's population grows at a decreasing rate due to the cube root. By around 2013, both populations are equal. After this point, Calcfield's population continues to grow faster than Mathville's.
The functions intersect at approximately t≈13.12 t \approx 13.12 , representing the year 2013. At this point, both cities have the same population.

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