Rosie is building two dog houses, one for each of her dogs. One dog is bigger than the other, so she decides to make one house the size of the other. How much smaller will the surface area of the small house be compared to that of the big house? Write your response as a fraction. (1 point)

Question

The surface area of the small house will be ____ the size of the big house.

Answer 1

To determine how much smaller the surface area of the small dog house is compared to the big dog house, we first need to establish the relationship between their sizes.

Let's denote:

• \(S\) as the surface area of the small dog house,
• \(B\) as the surface area of the big dog house.

If the small house is the same size as the big house, it means their surface areas are equal, which is:

\[ S = B \]

If we are interested in how much smaller the surface area of the small house is compared to the big house, since they are equal, we can express this as:

\[ S - B = 0 \]

Thus, the surface area of the small house is \(0\) times the size of the big house, which simplifies the fraction of the surface areas between them.

In terms of a fraction, we can write:

\[ \text{The surface area of the small house will be } \frac{S}{B} = \frac{B}{B} = 1 \]

This means the surface area of the small house is the same size as the big house, so there is no difference.

However, if we were to denote that the small house is proportionally smaller in size, typically in a scenario, it could be indicated as a fraction of their sizes depending on the specific dimensions.

If no additional detail is provided about their respective sizes, we typically conclude that the surface area of the small house holds equal proportions but can affirmatively state:

The surface area of the small house will be 1 times the size of the big house. If the argument indicates it is smaller than a specified dimension, it would depend on the calculations from the mentioned proportions which weren't defined in the above context.

Therefore, formally restating with proper analysis reflects:

The surface area of the small house will be \( \frac{0}{B} = 0 \) smaller than the big house, meaning ultimately it is not smaller but the same in the above context— thus:

The surface area of the small dog house will be 0 less than the big house if sizes are identical, leading to, as sought:

The surface area of the small house will be 0 the size of the big house.

Answer 2

wrong

Answer 3

I apologize for the confusion earlier. Let's clarify the situation.

If Rosie is building two dog houses and one is smaller than the other, we need to compare their surface areas to determine how much smaller the surface area of the small house is compared to that of the big house.

Let’s assume:

• The surface area of the big house is \( A_B \).
• The surface area of the small house is \( A_S \).

If the small house is proportionately smaller compared to the big house (for example, if it is half the size in all dimensions), we can derive that the areas would relate in a specific manner.

However, without specific measurements or ratios, I will present a general case. Assuming the small dog house is half the size of the big house in each dimension, the relationship between surface areas can be examined as follows:

If the dimensions of the bigger house are \( l, w, h \) (length, width, height), then the surface area \( A_B \) can be calculated as: \[ A_B = 2(lw + lh + wh) \]

If the smaller house's dimensions are \( \frac{l}{k}, \frac{w}{k}, \frac{h}{k} \) (where \( k \) is a scaling factor, e.g., 2 for half the size), then its surface area \( A_S \) becomes: \[ A_S = 2\left(\frac{l}{k}\frac{w}{k} + \frac{l}{k}\frac{h}{k} + \frac{w}{k}\frac{h}{k}\right) = \frac{2}{k^2}(lw + lh + wh) = \frac{A_B}{k^2} \]

If, for example, \( k = 2 \): \[ A_S = \frac{A_B}{2^2} = \frac{A_B}{4} \]

This means the surface area of the smaller house is \(\frac{1}{4}\) the size of the big house, indicating the small house is \(\frac{3}{4}\) smaller than the big house.

Thus, the correct expression representing the surface area of the small house compared to the big house, if the smaller house is half the size in each dimension, would be:

The surface area of the small house will be \(\frac{1}{4}\) the size of the big house.

If additional dimensions or ratios were specified, please provide them for a more precise comparison!

Answer 4

right

Answer 5

Great! I'm glad we clarified that. To summarize:

If the small dog house is proportionally smaller by a factor (like half the size in each dimension), the surface area of the small dog house is \(\frac{1}{4}\) the surface area of the big dog house. This means the small house's surface area is \(\frac{3}{4}\) smaller than that of the big house.

If you have any further questions or need help with anything else, feel free to ask!