To solve the problem of finding the area of the actual garden, we need to understand how scaling affects area.
The scale factor given is 3, which means the dimensions of the actual garden are 3 times the dimensions of the scale model. Since we are dealing with area, scaling affects the area by the square of the scale factor.
Let's go through the expressions one-by-one to see which one correctly represents the area of the actual garden.
1. [tex]$\frac{15}{3^2}$[/tex]: This expression suggests that we are dividing the model area by the square of the scale factor. However, this would give us a smaller area, which doesn’t make sense since the actual garden should be larger than the model.
2. [tex]$\frac{15}{3}$[/tex]: This expression suggests that we are dividing the model area by the scale factor. This also would give us a smaller area and doesn't match the problem's requirement.
3. [tex]$15 \times 3$[/tex]: This expression suggests that we are multiplying the model area by the scale factor once. Because area is a two-dimensional measurement, we must take into account the square of the scaling factor, not just the factor itself.
4. [tex]$15 \times 3^2$[/tex]: This expression correctly represents finding the area of the actual garden. Here, we are multiplying the model area by the square of the scale factor, which is how area scales when we change the dimensions by a factor.
The correct expression to find the area of the actual garden is:
[tex]\[ 15 \times 3^2 \][/tex]
Now, let's compute this to confirm:
[tex]\[ 3^2 = 3 \times 3 = 9 \][/tex]
[tex]\[ 15 \times 9 = 135 \][/tex]
Thus, the area of the actual garden is [tex]\( 135 \)[/tex] square feet. Hence, the expression [tex]$15 \times 3^2$[/tex] is correct.
