Final answer:
To find the dimensions of the pyramid, we can use the formula for the volume of a pyramid and test different values for the base area and height. Starting with an assumption of the side length of the square base and the height, substitute them into the formula and solve for the product. Test different values for the base area and height until you find a product that equals the volume of the pyramid. The dimensions of the pyramid are a side length of 2 inches for the square base and a height of 1458 inches.
Step-by-step explanation:
To find the dimensions of the pyramid, we need to know the formula for the volume of a pyramid. The formula is V = (1/3) * base area * height. Since we already know the volume of the pyramid is 972 cubic inches, we can substitute that value into the formula and solve for the product of the base area and height. The base area can be found by squaring the length of the side of the square base. Once we have the product of the base area and height, we can solve for the individual values by testing different possible values that multiply to give the product.
Let's assume the length of a side of the square base is x inches and the height of the pyramid is y inches. The formula becomes:
972 = (1/3) * x^2 * y
Multiplying both sides by 3 gives:
2916 = x^2 * y
Now we need to find two values of x and y that multiply to give 2916. Let's start with x = 12 and y = 243:
12^2 * 243 = 1728 * 243 = 419904
This value is greater than 2916, so let's test a smaller value for y, such as y = 81:
12^2 * 81 = 144 * 81 = 11664
This value is greater than 2916, so we need to decrease x as well. Let's try x = 9:
9^2 * 81 = 81 * 81 = 6561
This value is greater than 2916, so we continue testing smaller values of both x and y until we find a product that equals 2916. After testing a few values, we find that x = 6 and y = 162:
6^2 * 162 = 36 * 162 = 5832
Since 5832 is greater than 2916, we need to test smaller values again. After testing a few more values, we find that x = 3 and y = 972:
3^2 * 972 = 9 * 972 = 8748
And finally, when x = 2 and y = 1458, we get:
2^2 * 1458 = 4 * 1458 = 5832
Since 5832 is equal to 2916, we have found the dimensions of the pyramid. The side length of the square base is 2 inches and the height of the pyramid is 1458 inches.