Question

Best Pools Ever is building a new community pool in Harlem with the following features: • The pool is surrounded by a 2-foot-wide walkway. • The width of the pool is 23 ft less than the length of the pool. • The total area enclosed by the outer edge of the walkway is 1674 ft 2 . Which quadratic function in standard form models the area of the pool based on the length? A. l 2 – 15l – 76 = 0 B. l 2+ 8l – 1658 = 0 C. l 2– 15l – 1750 = 0 D. l 2+ 31l – 1566 = 0

Answer 1

To find the quadratic function that models the area of the pool based on the length \( l \), we can start by defining some variables.

Let:
- \( l \) = the length of the pool in feet
- The width of the pool is then \( w = l - 23 \) (since the width is 23 feet less than the length).

The walkway is 2 feet wide, so it adds to the dimensions of the pool. The external dimensions of the area that includes the walkway would therefore be:
- Length: \( l + 2 + 2 = l + 4 \) (2 feet on each side)
- Width: \( (l - 23) + 2 + 2 = l - 19 \)

The total area enclosed by the outer edge of the walkway is given as 1674 square feet. Thus, we can write the equation for the total area as:

\[
(l + 4)(l - 19) = 1674
\]

Expanding this equation:

\[
l^2 - 19l + 4l - 76 = 1674
\]
\[
l^2 - 15l - 76 = 1674
\]

Now, moving 1674 to the left side:

\[
l^2 - 15l - 76 - 1674 = 0
\]
\[
l^2 - 15l - 1750 = 0
\]

Thus, the quadratic function in standard form that models the area of the pool based on the length \( l \) is:

\[
l^2 - 15l - 1750 = 0
\]

So the correct choice is **C. \( l^2 - 15l - 1750 = 0 \)**.