Question

The beach area adjacent to a resort is (aprrox) a function of the ocean's distance from the resort as the tide comes in and out, and the tide's distance from the resort is a function of time. Let A(x) = 200x represent the beach area, and d(x) = -2^2+12x+10 represent the ocean's distance from the resort after x hours.
1.) Circle the dipposite function described above
a. A*d=A(d(x)) or b. d*A=d(A((x))
2.) Find the area of the beach after to hours

Answer 1

1.) To determine the appropriate function representation for the beach area in terms of the ocean's distance, we identify how the functions relate to each other. The beach area \( A \) is a function of the ocean's distance \( d \). Therefore, we want to use the function \( A(d(x)) \), which indicates that we are first calculating \( d(x) \) and then using that distance in the area function.

Hence, the correct option is:
**a. A*d=A(d(x))**

2.) To find the area of the beach after \( x = 2 \) hours, we first need to compute \( d(2) \) using the equation for the ocean's distance:

\[
d(x) = -2^2 + 12x + 10
\]

Substituting \( x = 2 \):

\[
d(2) = -2(2)^2 + 12(2) + 10
\]
\[
d(2) = -2(4) + 24 + 10
\]
\[
d(2) = -8 + 24 + 10
\]
\[
d(2) = 26
\]

Now that we have \( d(2) = 26 \), we can find the area of the beach \( A(26) \):

\[
A(d(2)) = A(26) = 200 \cdot 26
\]
\[
A(26) = 5200
\]

Thus, the area of the beach after 2 hours is \( \boxed{5200} \) square units.