Find the missing numbers for the dimensions and measures of a rectangular prism...

Length 5 m
Width ?
Height ?
Surface Area 94 m to the second power
Volume 60 m to the third power

Please answer the one's with a ?. Thank you.

we know that

w*h*5 = 60
2*5*w + 2*5*h + 2*w*h = 94
since w = 12/h, that gives

10(12/h) + 10h + 2(12/h)(h) = 94
120 + 10h^2 + 24h = 94h
10h^2 - 70h + 120 = 0
h^2-7h+12=0
(h-3)(h-4) = 0
h is 3 or 4, so
w is 4 or 3

The prism is 3x4x5

To find the missing dimensions and measures of a rectangular prism, we'll use the given information and equations for surface area and volume.

1. Given:
Length = 5 m
Width = ?
Height = ?
Surface Area = 94 m²
Volume = 60 m³

2. Surface Area of a Rectangular Prism:
The surface area of a rectangular prism can be calculated using the formula:
Surface Area = 2lw + 2lh + 2wh
where l is the length, w is the width, and h is the height.

3. Substitute the given values into the surface area formula:
94 = 2(5)(w) + 2(5)(h) + 2(w)(h)
Simplify: 94 = 10w + 10h + 2wh

4. Volume of a Rectangular Prism:
The volume of a rectangular prism is given by:
Volume = length × width × height

5. Substitute the given volume value into the volume formula:
60 = 5w × h

Now we have two equations:
94 = 10w + 10h + 2wh (Equation 1)
60 = 5w × h (Equation 2)

We can now solve these equations simultaneously to find the missing values.

6. Rearrange Equation 2 to solve for h:
h = 60 / (5w)

7. Substitute the value of h from Equation 6 into Equation 1:
94 = 10w + 10(60 / 5w) + 2w(60 / (5w))

Now simplify and solve for w:

94 = 10w + 120 / w + 24
Multiply both sides by w to get rid of the denominator:
94w = 10w² + 120 + 24w
Rearrange to form a quadratic equation:
10w² + 24w + 94w - 94w - 120 = 0
10w² + 24w - 120 = 0

8. Solve the quadratic equation using factoring, completing the square, or quadratic formula. In this case, we can factor it by taking out a common factor:
2(5w² + 12w - 60) = 0

Solve for w using: 5w² + 12w - 60 = 0

If we factor further, we get:
(5w - 10)(w + 6) = 0

So, either 5w - 10 = 0 or w + 6 = 0
Solve each equation separately:
5w - 10 = 0 --> 5w = 10 --> w = 2
w + 6 = 0 --> w = -6 (Discard this negative value since it represents the width)

Therefore, the width is 2 m.

9. Substitute the value of w = 2 into Equation 2 to find h:
60 = 5(2) × h
60 = 10h
h = 60 / 10
h = 6

Therefore, the width is 2 m and the height is 6 m.