To find the height of the rectangular prism, we start by understanding that the area of the cross section perpendicular to the base is given as 45 square inches. The cross-section area in a rectangular prism is calculated by multiplying the base area by the height of the prism.
First, let's calculate the base area of the prism. The base of the prism is a rectangle with:
- Length = 5 inches
- Width = 5 inches
The area of the base (A_base) is given by:
[tex]\[ A_\text{base} = \text{length} \times \text{width} \][/tex]
[tex]\[ A_\text{base} = 5 \, \text{inches} \times 5 \, \text{inches} = 25 \, \text{square inches} \][/tex]
Next, we know the cross-section area (A_cross-section) is 45 square inches. The cross-section area is also the product of the base area and the height of the prism (h):
[tex]\[ A_\text{cross-section} = A_\text{base} \times h \][/tex]
Substituting the given values:
[tex]\[ 45 \, \text{square inches} = 25 \, \text{square inches} \times h \][/tex]
To find the height [tex]\( h \)[/tex], we solve for [tex]\( h \)[/tex] by dividing both sides of the equation by the base area:
[tex]\[ h = \frac{45 \, \text{square inches}}{25 \, \text{square inches}} \][/tex]
[tex]\[ h = 1.8 \][/tex]
Therefore, the height of the prism is [tex]\( 1.8 \)[/tex] inches.
