Answer:
4.7 feet
Step-by-step explanation:
To find the height of the trapezoidal part of the prism, we first need to determine the area of the trapezoid's cross-section.
Then, we can use the volume formula for a prism to solve for the height.
The area [tex] A [/tex] of a trapezoid is given by the formula:
[tex] A = \dfrac{1}{2} (a + b) \times h [/tex]
where
- [tex] a [/tex] and [tex] b [/tex] are the lengths of the two parallel sides, and
- [tex] h [/tex] is the height of the trapezoid.
Given:
- Length = 14 ft
- Volume of the prism = 496.79 ft³ (Note: Volume units should be in cube)
- Side 1 of trapezoid = 6.2 ft
- Side 2 of trapezoid = 8.9 ft
First, let's find the area of the trapezoidal cross-section:
[tex] A = \dfrac{1}{2} (6.2 + 8.9) \times h [/tex]
[tex] A = \dfrac{1}{2} (15.1) \times h [/tex]
[tex] A = 7.55h [/tex]
Now, let's find the area of the trapezoidal prism:
[tex] \text{Volume} = \text{Area of trapezoid} \times \text{Length} [/tex]
[tex] 496.79 \text{ ft}^3 = 7.55h \times 14 \text{ ft} [/tex]
Now, we solve for [tex] h [/tex]:
[tex] 496.79 = 7.55h \times 14 [/tex]
[tex] h = \dfrac{496.79}{7.55 \times 14} [/tex]
[tex] h \approx \dfrac{496.79}{105.7} [/tex]
[tex] h \approx 4.7 \text{ ft} [/tex]
Therefore, the height of the prism is approximately 4.7 feet.
