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The radius of a circular pond is increasing at a constant rate, which can be modeled by the function $r(t)=5 t$, where $t$ is time in months. The area of the pond is modeled by the function $A(r)=\pi r^2$. The area of the pond with respect to time can be modeled by the composition $A(r(t))$.



Which function represents the area with respect to time?


A$A(r(t))=5 \pi r^2$
B$A(r(t))=5 \pi t^2$
C$A(r(t))=10 \pi r^2$
D$A(r(t))=25 \pi t^2$

Verified Answer :

  • Identify the functions \(r(t) = 5t\) and \(A(r) = \pi r^2\).
  • Compose the functions by substituting \(r(t)\) for \(r\) in \(A(r)\), resulting in \(A(r(t)) = \pi (5t)^2\).
  • Simplify the expression: \(A(r(t)) = 25 \pi t^2\).
  • The correct function is \(\boxed{A(r(t)) = 25 \pi t^2}\).

Explanation

  1. Analysis of the Problem

The problem defines the radius of a circular pond as a function of time \(t\) in months, given by \(r(t) = 5t\). The area of the pond as a function of its radius is given by \(A(r) = \pi r^2\). We need to determine the area of the pond as a function of time \(t\) by finding the composition of these two functions, denoted as \(A(r(t))\).

  1. Composing the Functions

To find the composition \(A(r(t))\), we substitute the expression for \(r(t)\) into the area function \(A(r)\). Since \(r(t) = 5t\), we replace the variable \(r\) in \(A(r) = \pi r^2\) with \(5t\):
\[A(r(t)) = A(5t) = \pi (5t)^2\]

  1. Simplification and Final Result

Now we simplify the expression \(\pi (5t)^2\). According to the rules of exponents, \((ab)^n = a^n b^n\). Applying this to \((5t)^2\):
\[(5t)^2 = 5^2 \cdot t^2 = 25t^2\]
Therefore, the full expression becomes:
\[A(r(t)) = \pi \cdot 25t^2 = 25 \pi t^2\]

Examples

Understanding composite functions is essential in modeling real-world dynamics where one variable depends on another, which in turn depends on a third. For instance, in biology, you might model the population of a certain bacteria colony that relies on the amount of available nutrients, where the nutrients are themselves depleting over time. By composing these functions, scientists can directly predict how the population changes over time based on the initial nutrient supply.

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