Question
Colette is putting a mat of width 3w inches and a frame of width w inches around a 16 inch by 48 inch poster. Find an expression for the amount of frame materials she needs
Answers
GPT-4o mini
To find an expression for the amount of frame materials Colette needs, let's first determine the dimensions of the poster and the frame around it.
The dimensions of the poster are:
- Width = 16 inches
- Height = 48 inches
Colette puts a mat around the poster, which adds a width of 3w inches to each side of the poster. The frame adds a width of w inches around the mat. Therefore, we need to calculate the total dimensions of the poster, mat, and frame.
1. **Calculate the dimensions including the mat:**
- The width of the poster plus the mat on both sides:
\[
16 + 2(3w) = 16 + 6w
\]
- The height of the poster plus the mat on both sides:
\[
48 + 2(3w) = 48 + 6w
\]
2. **Calculate the dimensions including the frame:**
- The width including the frame on both sides:
\[
(16 + 6w) + 2w = 16 + 6w + 2w = 16 + 8w
\]
- The height including the frame on both sides:
\[
(48 + 6w) + 2w = 48 + 6w + 2w = 48 + 8w
\]
3. **Total dimensions of the complete artwork (poster + mat + frame):**
- Total Width = \( 16 + 8w \)
- Total Height = \( 48 + 8w \)
4. **Calculate the area of the frame material needed:**
We find the area of the rectangle that would contain both the mat and the frame, and then subtract the area of the original poster.
- Area of the complete artwork (total area):
\[
\text{Area}_{\text{total}} = (16 + 8w)(48 + 8w)
\]
- Area of the poster:
\[
\text{Area}_{\text{poster}} = 16 \times 48 = 768 \text{ square inches}
\]
5. **Calculate the area of the frame material needed:**
\[
\text{Area}_{\text{frame}} = \text{Area}_{\text{total}} - \text{Area}_{\text{poster}}
\]
Calculating \(\text{Area}_{\text{total}}\):
\[
\text{Area}_{\text{total}} = (16 + 8w)(48 + 8w) = 16 \cdot 48 + 16 \cdot 8w + 48 \cdot 8w + 8w \cdot 8w
\]
\[
= 768 + 128w + 384w + 64w^2
\]
\[
= 768 + 512w + 64w^2
\]
So, the area of the frame material is:
\[
\text{Area}_{\text{frame}} = (768 + 512w + 64w^2) - 768 = 512w + 64w^2
\]
Thus, the expression for the amount of frame materials Colette needs is:
\[
\boxed{512w + 64w^2}
\]
The dimensions of the poster are:
- Width = 16 inches
- Height = 48 inches
Colette puts a mat around the poster, which adds a width of 3w inches to each side of the poster. The frame adds a width of w inches around the mat. Therefore, we need to calculate the total dimensions of the poster, mat, and frame.
1. **Calculate the dimensions including the mat:**
- The width of the poster plus the mat on both sides:
\[
16 + 2(3w) = 16 + 6w
\]
- The height of the poster plus the mat on both sides:
\[
48 + 2(3w) = 48 + 6w
\]
2. **Calculate the dimensions including the frame:**
- The width including the frame on both sides:
\[
(16 + 6w) + 2w = 16 + 6w + 2w = 16 + 8w
\]
- The height including the frame on both sides:
\[
(48 + 6w) + 2w = 48 + 6w + 2w = 48 + 8w
\]
3. **Total dimensions of the complete artwork (poster + mat + frame):**
- Total Width = \( 16 + 8w \)
- Total Height = \( 48 + 8w \)
4. **Calculate the area of the frame material needed:**
We find the area of the rectangle that would contain both the mat and the frame, and then subtract the area of the original poster.
- Area of the complete artwork (total area):
\[
\text{Area}_{\text{total}} = (16 + 8w)(48 + 8w)
\]
- Area of the poster:
\[
\text{Area}_{\text{poster}} = 16 \times 48 = 768 \text{ square inches}
\]
5. **Calculate the area of the frame material needed:**
\[
\text{Area}_{\text{frame}} = \text{Area}_{\text{total}} - \text{Area}_{\text{poster}}
\]
Calculating \(\text{Area}_{\text{total}}\):
\[
\text{Area}_{\text{total}} = (16 + 8w)(48 + 8w) = 16 \cdot 48 + 16 \cdot 8w + 48 \cdot 8w + 8w \cdot 8w
\]
\[
= 768 + 128w + 384w + 64w^2
\]
\[
= 768 + 512w + 64w^2
\]
So, the area of the frame material is:
\[
\text{Area}_{\text{frame}} = (768 + 512w + 64w^2) - 768 = 512w + 64w^2
\]
Thus, the expression for the amount of frame materials Colette needs is:
\[
\boxed{512w + 64w^2}
\]