Area of a rhombus is 72 ft. and the product of the diagonals is 144 ft. What is the length of each diagonal?

To find the length of each diagonal of a rhombus, we can use the formulas for the area and product of the diagonals.

The formula for the area of a rhombus is given by:
Area = (d1 * d2) / 2,
where d1 and d2 are the lengths of the diagonals.

Given that the area of the rhombus is 72 ft, we can substitute this value and rearrange the formula to solve for the product of the diagonals:

72 = (d1 * d2) / 2
144 = d1 * d2

Now we have a system of two equations:

72 = (d1 * d2) / 2
144 = d1 * d2

To solve this system, we can apply algebraic manipulation. We can multiply the first equation by 2 to eliminate the fraction:

144 = d1 * d2

Now we have the same equation for the product of the diagonals, which means both d1 and d2 must be factors of 144.

To find the factors of 144, we can list all the possible pairs of numbers that multiply to give 144:

1 * 144
2 * 72
3 * 48
4 * 36
6 * 24
8 * 18
9 * 16
12 * 12

From these pairs, we can see that the only pair that gives us a product of 144 and also satisfies the second equation is 9 * 16.

Therefore, the length of each diagonal is 9 ft and 16 ft.