Answer:
The length of the base in feet is 8 ft.
Explanation:
The volume V of the rectangular prism is given by
V = l × w × h
But Area, A = l × w
∴ V = A × h
Where A is the base area and h is the height
From the question
A = 56 ft.² and V = 840 ft.³
∴ 840 ft.³ = 56 ft.² × h
840 ft.³ / 56 ft.² = h
15 ft. = h
∴ h = 15 ft.
Hence, the height of the rectangular prism is 15 ft.
Also, from the question
the sum of the length and the width of the base is equal to the height
i.e l + w = h
where l is the length of the base and w is the width of the base
Then, l + w = 15 ------ (1)
Also, A = l × w
Then, 56 = l × w ------- (2)
From equation (1)
l + w = 15
Then, w = 15 - l ------- (3)
Substitute this into equation (2)
56 = l × w
56 = l × (15-l)
56 = 15l - l²
Then,
l²- 15l + 56 = 0
l² -8l -7l + 56 = 0
l(l - 8) -7(l - 8) = 0
(l - 7)(l - 8) = 0
l - 7 = 0 OR l - 8 = 0
∴ l = 7 OR l = 8
Substitute the values of l into equation (3)
w = 15 - l
when l = 7
w = 15 - 7
w = 8
and when l = 8
w = 15 - 8
w = 7
From the question, the length of the base is longer than the width
∴ l = 8 and w = 7
Hence, the length of the base in feet is 8 ft.