Question

edgar builds a sand castle with a rectangular base. the side lengths of the base are 25in and 16in. he wants to surround the castle's base with a moat that is w inches wide. write a quadratic function in standard form to represent the combined area taken up by the castle and the moat.

Answer 1

To find the combined area taken up by Edgar's sand castle and the moat, we'll start by determining the dimensions of the area that includes both the sand castle and the moat.

1. **Dimensions of the castle**: The base dimensions of the sand castle are:
- Length = 25 inches
- Width = 16 inches

2. **Moat width**: The moat is \( w \) inches wide, which surrounds the castle on all sides.

3. **Dimensions including the moat**:
- The total length of the area covered by the castle and the moat will be \( 25 + 2w \) (adding \( w \) on both ends).
- The total width of the area will be \( 16 + 2w \) (again, adding \( w \) on both sides).

4. **Area of the castle and moat**: The area \( A \) of the rectangle that includes both the castle and the moat can be represented as:
\[
A = \text{length} \times \text{width} = (25 + 2w)(16 + 2w)
\]

5. **Expanding the equation**:
\[
A = (25 + 2w)(16 + 2w)
\]
Using the distributive property (FOIL method):
\[
A = 25 \cdot 16 + 25 \cdot 2w + 16 \cdot 2w + 2w \cdot 2w
\]
\[
A = 400 + 50w + 32w + 4w^2
\]
Combine like terms:
\[
A = 400 + 82w + 4w^2
\]

6. **Rearranging to standard form**: The standard form of a quadratic function is:
\[
A = ax^2 + bx + c
\]
Thus, we organize it as:
\[
A = 4w^2 + 82w + 400
\]

Therefore, the quadratic function that represents the combined area taken up by the castle and the moat is:
\[
A(w) = 4w^2 + 82w + 400
\]