How do I determine the length and width of a prism that has a height of 3 in, all edges are integers, the numerical values of the volume and of the surface area are equal and the volume is as small as possible?
To determine the length and width of the prism, we need to understand the properties of a rectangular prism and how to calculate its volume and surface area.
A rectangular prism has six rectangular faces, and each pair of opposite faces has equal dimensions. The volume of a rectangular prism is given by the formula V = lwh, where V represents the volume, l represents the length, w represents the width, and h represents the height.
The surface area of a rectangular prism can be calculated using the formula SA = 2lw + 2lh + 2wh, where SA represents the surface area.
In this case, we are given that the prism's height is 3 inches, and the numerical values of the volume and surface area are equal. To minimize the volume, we need to find the smallest possible values for the length and width while still satisfying the conditions.
To begin, let's assign variables to the length (l) and width (w). Since all edges are integers, we can assume that both l and w are positive integers.
Let's substitute these variables into the formulas:
Volume: V = lwh
Surface Area: SA = 2lw + 2lh + 2wh
Since the numerical values of the volume and surface area are equal, we can set up the following equation:
2lw + 2lh + 2wh = lwh
Next, we simplify the equation and rearrange it:
2lw + 2lh + 2wh - lwh = 0
lwh - 2lw - 2lh - 2wh = 0
lwh - 2lw - 2lh - 2wh + 4lw + 6lh + 6wh = 4lw + 6lh + 6wh
lwh + 4lw + 6lh + 6wh = 4lw + 6lh + 6wh
lwh + 4lw + 6lh + 6wh - 4lw - 6lh - 6wh = 0
lwh - 4lw - 6lh = 0
l(wh - 4w - 6h) = 0
From this equation, we can deduce that either l = 0 or (wh - 4w - 6h) = 0.
Since the length and width should be positive integers, we can conclude that l cannot be zero. Thus, we are left with the equation:
wh - 4w - 6h = 0
To minimize the volume, we need to find the smallest possible values for w and h that satisfy this equation.
One way to approach this is by factoring the equation. We can think of it as a quadratic equation in terms of w, where the coefficient of w² is 1:
w(h - 4) - 6h = 0
To factor this equation, we look for two numbers, p and q, whose product is equal to the product of the coefficient of w² (1) and the constant term (-6h). In this case, it is -6h. Additionally, we want the sum of these two numbers (p + q) to be equal to the coefficient of w (-4).
After determining the factors, we can rewrite the equation:
w(h - 4) - 6h = 0
w(h - 4) = 6h
Now we have two possible cases:
Case 1: (h - 4) = 1
By setting (h - 4) equal to 1, we can solve for h:
h - 4 = 1
h = 1 + 4
h = 5
Now that we have the value of h, we can substitute it back into the equation to find w:
w(5 - 4) = 6(5)
w = 30/1
w = 30
Thus, in this case, the values of length (l), width (w), and height (h) are 30, 30, and 5, respectively.
Case 2: (h - 4) = 2
By setting (h - 4) equal to 2, we can solve for h:
h - 4 = 2
h = 2 + 4
h = 6
Now that we have the value of h, we can substitute it back into the equation to find w:
w(6 - 4) = 6(6)
2w = 36
w = 36/2
w = 18
Thus, in this case, the values of length (l), width (w), and height (h) are 18, 18, and 6, respectively.
Therefore, you have two possible sets of values for the length and width of the prism: (30, 30) and (18, 18). These sets of values satisfy the conditions of the height being 3 inches, all edges being integers, and the volume being as small as possible while still having the numerical value of the volume and surface area equal.