Solve the equation you write in Part B by completing the square.
A
a
6x^2+28x
B
b
6x^2+28x=384
C
c
13 in. * 6in.* 6in.
Practice makes perfect
a
We obtain the shape below when we open the faces of the prism.
There are 4 rectangles with an area of x(x+7) and 2 squares with an area of x^2. Therefore, we can write the surface area of the prism A_p as follows.
A_p= 4* x(x+7) + 2* x^2
Let's simplify the expression.
The surface area is equal to 6x^2+28x.
A_p=4* x(x+7) +2* x^2
⇕
A_p=6x^2+28x
b
The surface area A_c of a cube is equal to six times the area of one of its faces. We have a cube with edges 8inches long. Then the surface area of it is equal to 6* 8^2.
A_c=6* a^2 ⇒ A_c=6 * 8^2
Since the prism and the cube has the same surface area, the expression in Part A is equal to 6* 8^2= 384
A_p= A_c ⇔
6x^2+28x= 384
c
We will solve the quadratic equation we wrote in Part B by completing the square. 6x^2+28x=384
Let's divide each side by 6 so the coefficientof x^2 will be 1.
The solutions for this equation are x=- 73± 253. Let's separate them into the positive and negative cases.
x=- 7/3± 25/3
x_1=- 7/3+ 25/3
x_2=- 7/3- 25/3
x_1 =6
x_2≈ - 10.6
The only reasonable solution is 6 as lengths cannot be negative. The value of x is 6 inches. Therefore, the dimensions of the prism are 6, 6, and 13.
13 in. * 6in.* 6in.
