The surface area of a right rectangular prism is 112 sq units. The height is twice the width. The length is 4 units more than the height . Find the volume.
width --- x
length --- x+4
height --- 2x
surface area
= 2(x)(x+4) + 2(x)(2x) + 2(2x)(x+4)
= 112
2x^2 + 8x + 4x^2 + 4x^2 + 16x = 112
10x^2 + 24x - 112=0
5x^2 + 12x - 56 = 0
x = (-12 ± √1264)/10
= 2.355 or a negative
width = 2.355
length = 6.355
height = 4.711
check:
2(2.355(6.355) + 2(2.355)(4.711) + 2(6.355)(4.711) = 112
volume = (2.355)(6.355)(4.711) = .....
To find the volume of a right rectangular prism, we need to know the formula for the volume of a rectangular prism, which is V = lwh, where l represents the length, w represents the width, and h represents the height.
Let's break down the information given in the problem:
- The surface area of the prism is 112 square units.
- The height is twice the width (h = 2w).
- The length is 4 units more than the height (l = h + 4).
To find the volume, we first need to find the values of l, w, and h that satisfy all the given conditions. Let's solve this step by step:
1. Find the value of w:
Since the height is twice the width, we can replace h in terms of w: h = 2w.
2. Substitute the value of h into the equation for length:
Since the length is 4 units more than the height, we can replace l in terms of w: l = h + 4 = 2w + 4.
3. Find the surface area:
The surface area of a rectangular prism is given by the formula 2lw + 2lh + 2wh. Substituting the values we have:
112 = 2(2w + 4)(2w) + 2(2w + 4)w + 2wh.
4. Simplify the equation:
Expand and simplify the equation:
112 = 8w^2 + 16w + 4w^2 + 8w + 4w + 8w^2 + 4w.
112 = 20w^2 + 40w.
5. Solve for w:
Rearrange the equation to set it equal to zero:
20w^2 + 40w - 112 = 0.
6. Solve the quadratic equation:
Now we can solve for w using factoring, completing the square, or the quadratic formula. In this case, we will use factoring.
Factor out 4:
4(5w^2 + 10w - 28) = 0.
Factor the remaining quadratic equation:
4(5w^2 + 14w - 4w - 28) = 0.
4(w(5w + 14) - 4(5w + 14)) = 0.
4(w - 4)(5w + 14) = 0.
Set each factor equal to zero:
w - 4 = 0 or 5w + 14 = 0.
Solve for w:
w = 4 or w = -14/5 (reject this solution because width can't be negative).
Therefore, the width (w) is 4 units.
7. Find the height (h) and length (l):
Since h = 2w, the height is 2(4) = 8 units.
Since l = h + 4, the length is 8 + 4 = 12 units.
So, the height (h) is 8 units and the length (l) is 12 units.
8. Calculate the volume:
Now that we have the values of length (l), width (w), and height (h), we can calculate the volume:
V = lwh
V = 12(4)(8)
V = 384 cubic units.
Therefore, the volume of the right rectangular prism is 384 cubic units.