Questions: Graphs and Functions Word problem involving composition of two functions The surface area S(r) (in square meters) of a spherical balloon with radius r meters is given by S(r)=4 π r^2. The radius P(t) (in meters) after t seconds is given by P(t)=8/3 t. Write a formula for the surface area N(t) (in square meters) of the balloon after t seconds. It is not necessary to simplify. N(t)=

Graphs and Functions
Word problem involving composition of two functions

The surface area S(r) (in square meters) of a spherical balloon with radius r meters is given by S(r)=4 π r^2.
The radius P(t) (in meters) after t seconds is given by P(t)=8/3 t.
Write a formula for the surface area N(t) (in square meters) of the balloon after t seconds.
It is not necessary to simplify.
N(t)=

Transcript text: Graphs and Functions Word problem involving composition of two functions The surface area $S(r)$ (in square meters) of a spherical balloon with radius $r$ meters is given by $S(r)=4 \pi r^{2}$. The radius $P(t)$ (in meters) after $t$ seconds is given by $P(t)=\frac{8}{3} t$. Write a formula for the surface area $N(t)$ (in square meters) of the balloon after $t$ seconds. It is not ne ${ }^{\text {ary to simplify. }}$ \[ N(t)= \]

Solution

Solution Steps

To find the surface area of the balloon as a function of time, we need to compose the two given functions. The surface area $ S(r) $ is a function of the radius $ r $, and the radius $ P(t) $ is a function of time $ t $. Therefore, the surface area as a function of time $ N(t) $ can be found by substituting $ P(t) $ into $ S(r) $.

Step 1: Define the Functions

The surface area $ S(r) $ of a spherical balloon with radius $ r $ is given by the formula: \[ S(r) = 4 \pi r^2 \]

The radius $ P(t) $ of the balloon as a function of time $ t $ is given by: \[ P(t) = \frac{8}{3} t \]

Step 2: Compose the Functions

To find the surface area as a function of time, substitute $ P(t) $ into $ S(r) $: \[ N(t) = S(P(t)) = 4 \pi \left(\frac{8}{3} t\right)^2 \]

Step 3: Simplify the Expression

Simplify the expression for $ N(t) $: \[ N(t) = 4 \pi \left(\frac{8}{3} t\right)^2 = 4 \pi \left(\frac{64}{9} t^2\right) = \frac{256}{9} \pi t^2 \]