The base of a rectangular prism has a length of 13 inches and a width of 1/2 inch. The volume of the prism is less than 65 cubic inches. Find all possible heights of the prism. Show all your work as an inequality.
THANK YOU!!
1-13
let the height be x
(13)(1/2)x < 65
x < 2(65)/13
x < 10 where 0 < x < 10 , x is a real number
of course there would be an infinite number of solutions if you allow decimals or fractions.
To find the possible heights of the rectangular prism, we can use the formula for the volume of a rectangular prism:
Volume = Length * Width * Height
Given:
Length = 13 inches
Width = 1/2 inch
Volume < 65 cubic inches
We can substitute these values into the formula and create an inequality. Let's solve it step by step:
Volume = Length * Width * Height
65 > 13 * (1/2) * Height [Since volume is less than 65, we use the greater than sign]
65 > (13/2) * Height [Multiply 13 and (1/2)]
65 > (13/2) * Height [Simplify]
Now, let's isolate the variable (Height) by dividing both sides of the inequality by (13/2):
(65 / (13/2)) > Height
Simplifying further:
(65 * 2) / 13 > Height
130 / 13 > Height
10 > Height
Therefore, the height of the prism must be less than 10 inches for the volume to be less than 65 cubic inches.
Inequality: Height < 10