I have a rectangular prism with a volume of 48 cubic feet and a height of 3/4 foot.
The base of the rectangular prism is not a square and the width is greater than 2 feet. What is the length and width of the rectangular prism?
What is the volume of the box pictured below?
A rectangular prism of length 3 and 1 over 2 feet, width 3 and 1 over 8 feet, and height 4 over 5 foot is shown.
To find the length and width of the rectangular prism, we need to use the formula for the volume of a rectangular prism: Volume = Length x Width x Height.
Given that the volume is 48 cubic feet and the height is 3/4 foot, we can substitute these values into the formula:
48 = Length x Width x (3/4)
To simplify the equation, we can multiply both sides by 4/3:
(4/3) x 48 = Length x Width
Multiplying 4/3 by 48 gives us:
64 = Length x Width
Now we need to find two factors of 64 that satisfy the conditions given: the width must be greater than 2 feet, and the base of the rectangular prism is not a square.
Let's list the factors of 64:
1, 2, 4, 8, 16, 32, 64
From the list, the only pair of factors that satisfies the conditions is 8 and 8. However, the base of the prism is not a square, so we cannot use this pair.
Another possibility is to use 32 and 2 as the length and width, respectively.
So, the length of the rectangular prism is 32 feet, and the width is 2 feet.
To find the length and width of the rectangular prism, we can use the formula for volume of a rectangular prism:
Volume = length * width * height
Given that the volume is 48 cubic feet and the height is 3/4 foot, we can substitute these values into the formula:
48 = length * width * (3/4)
To simplify, let's multiply both sides of the equation by 4:
192 = length * width * 3
Now divide both sides by 3:
64 = length * width
Now we need to find two numbers whose product is 64 and the width is greater than 2 feet. To do this, we can list the factors of 64:
1, 2, 4, 8, 16, 32, 64
Since the width is greater than 2 feet, we can eliminate the first two options. Looking at the remaining factors, we see that 8 and 8 give us a product of 64 and satisfies the condition. So, the length of the rectangular prism is 8 feet and the width is also 8 feet.
If the width is w, and the length is x, then
w*x*(3/4) = 48
wx = 64
The only integer factors of 64 greater than 2 are 4,8,16,32
So, the base of the prism must be
4x16
Of course, if integer values are not required, any number greater than 2 will work as the width, as long as the length is 64/w.