We will start finding the dimensions of the mold by recalling the formula for the volume of a pyramid.
V=Bh/3
In this formula, B is the area of the base and h is the height of the pyramid. According to the information on the diagram, the base is an x* x square and the height is x+1. Let's use these expressions to write the volume in terms of x.
V=x^2(x+1)/3
We are given that the volume is 4 cubic feet. Let's use this value to set up and simplify an equation for x.
According to the Rational Root Theorem, the integer solutions of this equation are factors of the constant term -12. Since x represents a length, we are only interested in positive solutions. Let's list the positive factors of -12.
1, 2, 3, 4, 6, 12
We will substitute these into the left hand side of the equation to see whether any of these numbers is a solution.
x
x^3+x^2-12
Value
1
1^3+ 1^2-12
- 10
2
2^3+ 2^2-12
0 âś“
We found that x=2 is a solution. Since the volume is increasing in x, there is only one x-value that gives the 4 cubic feet volume. This means that we do not need to look further, we found the only solution. Let's substitute x=2 into the measurements given on the diagram to find the dimensions of the mold.
Base:& 2feet* 2feet squared
Height:& 2+1=3feet
