Question

note: figure not drawn to scale.
she needs the enlargement to follow the scale of \\(\frac{1}{4}\\) of a foot to 3 feet. this will result in an image with a length of 8 feet and width of 6 feet.
what is the unit rate of area in square feet of the enlarged image per square foot of the original image?
a. 16 square feet per square foot
b. 48 square feet per square foot
c. 144 square feet per square foot
d. 32 square feet per square foot

Explanation:

Step1: Find the scale factor

The scale is \(\frac{1}{4}\) foot to 3 feet. To find the scale factor (ratio of enlarged length to original length), we calculate \(\frac{3}{\frac{1}{4}}\) = \(3\times4 = 12\). Wait, no, actually, the original length (from the figure) is \(\frac{2}{3}\) foot? Wait, maybe I misread. Wait, the problem says: "She needs the enlargement to follow the scale of \(\frac{1}{4}\) of a foot to 3 feet. This will result in an image with a length of 8 feet and width of 6 feet. What is the unit rate of the enlarged image per square foot of the original image?"

Wait, first, find the original length and width using the scale. The scale is \(\frac{1}{4}\) ft (original) : 3 ft (enlarged). So scale factor \(k=\frac{\text{enlarged length}}{\text{original length}}=\frac{3}{\frac{1}{4}} = 12\)? Wait, no, if the enlarged length is 8 ft, then original length \(L_o=\frac{8}{k}\), and from scale, \(\frac{1}{4}\) ft (original) corresponds to 3 ft (enlarged), so \(k = \frac{3}{\frac{1}{4}}=12\). Wait, but the enlarged length is 8 ft, so original length \(L_o=\frac{8}{12}=\frac{2}{3}\) ft. Similarly, enlarged width is 6 ft, so original width \(W_o=\frac{6}{12}=\frac{1}{2}\) ft? Wait, no, maybe the scale is \(\frac{1}{4}\) ft (original) to 3 ft (enlarged), so the ratio of enlarged to original is \(3\div\frac{1}{4}=12\). So the scale factor (enlarged/original) is 12.

Now, area of enlarged image: \(A_e = 8\times6 = 48\) square feet.

Area of original image: \(A_o=\text{original length}\times\text{original width}\). From the figure, the original length (from the purple square) is \(\frac{2}{3}\) ft? Wait, the figure has \(\frac{2}{3}\) ft. Wait, maybe the original length is \(\frac{8}{12}=\frac{2}{3}\) ft (since 8 ft enlarged, scale factor 12, so original \(8\div12=\frac{2}{3}\) ft), and original width is \(6\div12=\frac{1}{2}\) ft? Wait, no, maybe the original dimensions are found by the scale. Wait, the scale is \(\frac{1}{4}\) ft (original) : 3 ft (enlarged). So if the enlarged length is 8 ft, then original length \(L_o = 8\times\frac{\frac{1}{4}}{3}=8\times\frac{1}{12}=\frac{2}{3}\) ft. Similarly, enlarged width 6 ft, original width \(W_o = 6\times\frac{\frac{1}{4}}{3}=6\times\frac{1}{12}=\frac{1}{2}\) ft. Then original area \(A_o=\frac{2}{3}\times\frac{1}{2}=\frac{1}{3}\) square feet? Wait, that can't be. Wait, maybe I messed up the scale.

Wait, another approach: The scale is \(\frac{1}{4}\) ft (original) to 3 ft (enlarged), so the ratio of enlarged to original is \(3\) ft (enlarged) per \(\frac{1}{4}\) ft (original), so the linear scale factor \(k = \frac{3}{\frac{1}{4}} = 12\). So the area scale factor is \(k^2 = 12^2 = 144\). Wait, but the enlarged area is \(8\times6 = 48\) square feet. Then the unit rate (enlarged area per original area) is \(\frac{48}{A_o}\), where \(A_o\) is original area. But original area: if linear scale factor is 12, then original length is \(\frac{8}{12}=\frac{2}{3}\) ft, original width is \(\frac{6}{12}=\frac{1}{2}\) ft, so original area is \(\frac{2}{3}\times\frac{1}{2}=\frac{1}{3}\) sq ft. Then \(\frac{48}{\frac{1}{3}} = 48\times3 = 144\). Ah, that makes sense. So the unit rate is 144 square feet of enlarged image per square foot of original image.

Wait, let's re-express:

1. Find the scale factor (ratio of enlarged length to original length). The scale is \(\frac{1}{4}\) ft (original) : 3 ft (enlarged). So if the enlarged length is 8 ft, then original length \(L_o = 8 \times \frac{\frac{1}{4}}{3} = \frac{8}{12} = \frac{2}{3}\) ft. Similarly, enlarged width 6 ft, original width \(W_o = 6 \times \frac{\frac{1}{4}}{3} = \frac{6}{12} = \frac{1}{2}\) ft.

2. Calculate area of enlarged image: \(A_e = 8 \times 6 = 48\) sq ft.

3. Calculate area of original image: \(A_o = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}\) sq ft.

4. Unit rate (enlarged per original) is \(\frac{A_e}{A_o} = \frac{48}{\frac{1}{3}} = 48 \times 3 = 144\).

Alternatively, using scale factor for area: the linear scale factor \(k = \frac{\text{enlarged length}}{\text{original length}} = \frac{3}{\frac{1}{4}} = 12\) (from the scale: \(\frac{1}{4}\) ft original to 3 ft enlarged, so \(k = 3 / (1/4) = 12\)). Then area scale factor is \(k^2 = 12^2 = 144\). So the area of enlarged image is 144 times the area of original image. Therefore, the unit rate is 144 square feet of enlarged image per square foot of original image.

Step2: Verify

Enlarged area: 8*6=48. Original area: (8/12)*(6/12)= (2/3)*(1/2)=1/3. 48/(1/3)=144. So the unit rate is 144.

Answer:

C. 144 square feet per square foot