The height of a rectangular prism is 20 centimeters. It has a surface area of 2,400 square centimeters. What are two possible sets of lengths and widths? Find one set of dimensions with l and w equal in length as well as a set of dimensions that are not equal.
If the base has area B, then the prism's surface is
2B + 2*20(L+W)
If L=W, then
2L^2 + 80L = 2400
L=W=20
See what you can do with L not equal W.
To find the different measurements of length and width, start with an algebraic equation. Substitute the length or width with a reasonable number like thirty. Then simplify the equation to find the other measurement.
2(20×30+20w+30w) = 2,400
20×30+20w+30w = 1,200
600+50w = 1,200
50w = 600
w = 12
This means the width would be 12cm and the length would be 30cm. Just know this is an example of one possibility for an answer.
What even are the possible lengths ?
To find the possible sets of lengths and widths, we can use the formulas for surface area and volume of a rectangular prism, as well as the given information.
The formula for the surface area of a rectangular prism is:
SA = 2lw + 2lh + 2wh
Here, l, w, and h represent the length, width, and height of the prism, respectively.
Since the surface area is given as 2,400 square centimeters, we can set up the following equation:
2,400 = 2lw + 2lh + 2wh
We also know that the height is 20 centimeters. Now, we can proceed to solve for the possible sets of lengths and widths.
1. Set of dimensions with l and w equal in length:
Let's assume l = w = x (where x represents the common length and width).
We can rewrite the equation using this assumption:
2,400 = 2(x)(x) + 2(x)(20) + 2(20)(x)
2,400 = 4x^2 + 40x + 40x
2,400 = 4x^2 + 80x
Simplifying further:
2x^2 + 40x - 600 = 0
Factoring the equation:
2(x - 10)(x + 30) = 0
Setting each factor equal to zero:
x - 10 = 0 --> x = 10
x + 30 = 0 --> x = -30 (not a valid dimension)
Since we are dealing with dimensions, we disregard the negative value. Therefore, one possible set of dimensions, where l and w are equal, is:
Length (l) = Width (w) = 10 centimeters
2. Set of dimensions with l and w not equal in length:
We can choose a different set of dimensions where l and w are not equal. Let's assume l = 40 (length) and w = 15 (width), for example.
Now we substitute these values into the surface area equation:
2,400 = 2(40)(15) + 2(40)(20) + 2(20)(15)
2,400 = 1,200 + 1,600 + 600
2,400 = 3,400
This equation is not satisfied, meaning that l = 40 and w = 15 are not valid dimensions for a rectangular prism with a height of 20 centimeters and a surface area of 2,400 square centimeters.
In conclusion, one possible set of dimensions where the length and width are equal is 10 centimeters. However, there is no valid set of dimensions where the length and width are not equal.