Question

terry made a plastic box - house with the dimensions shown below. how many square inches of plastic did terry use before cutting the hole?

Explanation:

Step1: Identify box dimensions

The box has length $l = 10$ inches, width $w= 6$ inches, and height $h = 6$ inches.

Step2: Recall surface - area formula for rectangular prism

The surface - area formula of a rectangular prism is $SA = 2(lw+lh + wh)$.

Step3: Substitute values into formula

$SA=2(10\times6 + 10\times6+6\times6)$.
First, calculate the products inside the parentheses: $10\times6 = 60$, $10\times6 = 60$, and $6\times6 = 36$.
Then, $SA=2(60 + 60+36)$.
Next, add the numbers inside the parentheses: $60 + 60+36=156$.
Finally, $SA = 2\times156=312$ square inches. But it seems there is a mistake in our initial approach. Let's calculate the area of each face separately.
The box has two faces with area $10\times6 = 60$ square inches each, two faces with area $10\times6=60$ square inches each, and two faces with area $6\times6 = 36$ square inches each.
Total area $=2\times60+2\times60 + 2\times36=120+120 + 72=312$ square inches. However, if we assume the net of the box shown is correct and we calculate the area of each part of the net:
There are two $10\times6$ rectangles, two $6\times6$ rectangles and two more $10\times6$ rectangles.
Area of two $10\times6$ rectangles: $2\times(10\times6)=120$ square inches.
Area of two $6\times6$ rectangles: $2\times(6\times6) = 72$ square inches.
Another two $10\times6$ rectangles: $2\times(10\times6)=120$ square inches.
Total area $=120+72 + 120=312$ square inches. If we assume the correct values from the problem - setup and re - calculate the surface area of the rectangular prism using $SA=2(lw+lh+wh)=2(10\times6 + 10\times6+6\times6)=2(60 + 60+36)=2\times156 = 312$ square inches. But if we consider the problem in a different way and calculate the sum of the areas of the individual rectangles in the net:
We have three pairs of rectangles.
Pair 1: Two rectangles with dimensions $10\times6$: Area $A_1=2\times10\times6 = 120$ square inches.
Pair 2: Two rectangles with dimensions $10\times6$: Area $A_2=2\times10\times6=120$ square inches.
Pair 3: Two rectangles with dimensions $6\times6$: Area $A_3=2\times6\times6 = 72$ square inches.
Total area $A=A_1+A_2+A_3=120 + 120+72=312$ square inches.

Answer:

There is an error in the given options as the correct answer should be 312 square inches. But if we assume some mis - reading or wrong values in our calculation and re - check based on the formula for the surface area of a rectangular prism $SA = 2(lw+lh+wh)$ with $l = 10$, $w = 6$ and $h = 6$:
$SA=2(10\times6+10\times6 + 6\times6)=2(60+60 + 36)=2\times156=312$ square inches. If we consider the net of the box and sum up the areas of all the rectangles in the net:
We have:
Two rectangles of area $10\times6 = 60$ (total $2\times60 = 120$), two rectangles of area $10\times6=60$ (total $2\times60 = 120$) and two rectangles of area $6\times6 = 36$ (total $2\times36=72$).
Sum of areas $=120+120 + 72=312$ square inches. Since the options provided do not have 312 square inches, there might be a mis - understanding in the problem or in our interpretation. But based on the geometric calculation of the surface area of the rectangular prism formed by the given dimensions, the area of the plastic before cutting the hole should be 312 square inches.