Sean has a rectangular painting with an area of 80 square inches. He wants to enlarge the painting to 320 square inches.
If the length and width of the original painting are 10 inches and 8 inches, what will the dimensions of the enlarged painting be?

 **A.** 

12 inches and 10 inches

 **B.** 

14 inches and 12 inches

 **C.** 

20 inches and 16 inches

 **D.** 

40 inches and 32 inches

To find the dimensions of the enlarged painting, we need to understand the relationship between the original and enlarged areas. Given the original dimensions and the desired area, we can use the following steps:

1. **Determine the scale factor for the area:**

   The area of the original painting is \( 80 \) square inches.
   The area of the enlarged painting is \( 320 \) square inches.

   The scale factor for the area is the ratio of the enlarged area to the original area:
   \[
   \text{Scale factor for area} = \frac{\text{New area}}{\text{Original area}} = \frac{320}{80} = 4
   \]

2. **Determine the scale factor for the dimensions:**

   Since the area scales by the square of the linear dimensions, we take the square root of the area scale factor to find the linear scale factor:
   \[
   \text{Linear scale factor} = \sqrt{4} = 2
   \]

3. **Calculate the new dimensions:**

   Multiply the original dimensions by the linear scale factor:
   \[
   \text{New length} = \text{Original length} \times \text{Linear scale factor} = 10 \times 2 = 20 \, \text{inches}
   \]
   \[
   \text{New width} = \text{Original width} \times \text{Linear scale factor} = 8 \times 2 = 16 \, \text{inches}
   \]

The new dimensions of the enlarged painting are 20 inches by 16 inches.

Therefore, the correct answer is:
**C. 20 inches and 16 inches**

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. What would the dimensions be if Sean wanted to enlarge the painting to 640 square inches?
2. How does the perimeter of the original painting compare to the perimeter of the enlarged painting?
3. If the original painting's frame adds 1 inch to each side, what are the dimensions of the painting including the frame?
4. What is the aspect ratio of the original painting?
5. If the enlarged painting is framed with a 2-inch wide frame, what are the outer dimensions including the frame?
6. How much more material is needed to cover the enlarged painting compared to the original painting?
7. If the original painting is reduced to 40 square inches, what will the new dimensions be?
8. How would the dimensions change if the width is doubled while keeping the same area?

Tip: When scaling objects, keep in mind that linear dimensions scale by the square root of the area factor, while volumes scale by the cube root of the volume factor.

Explore how to calculate the dimensions of an enlarged painting when the area is increased from 80 square inches to 320 square inches. Understand the concept of scale factor and its application in geometry. Suitable for high school students and those interested in geometry and area scaling.

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