a rectangular swimming pool has a perimeter of 24 feet. what should the dimesions of the pool be to create a maximum area? what would this area be?

If x is the one side's dimension, the adjacent side's dimension is 12 - x. (That makes the perimeter 24). The area is then
A = x (12 - x) = 12 x - x^2
x can take values from 0 to 12

One can use calculus or graphical means to find that the largest value of A occurs when x = 6. This corresponds to a square pool. In that case, A = 36 square feet.

find 3 consecutive integers such that the product of the second and third intege

find 3 consecutive integers such that the product of the second and third intege

find 3 consecutive integers such that the product of the second and third intege

To find three consecutive integers such that the product of the second and third integers is equal to twice the first integer, you can follow these steps:

1. Assign variables to the integers. Let's assume the first integer is n.

2. The second consecutive integer would be n + 1, and the third consecutive integer would be n + 2.

3. Express the condition that the product of the second and third integers is equal to twice the value of the first integer as an equation:
(n + 1) * (n + 2) = 2n

4. Expand the equation:
n^2 + 3n + 2 = 2n

5. Simplify the equation and move all terms to one side to get a quadratic equation:
n^2 + n - 2 = 0

6. Factorize the quadratic equation:
(n + 2)(n - 1) = 0

7. Set each factor equal to zero and solve for n:
n + 2 = 0 --> n = -2
n - 1 = 0 --> n = 1

So, the two possible values for the first integer are -2 and 1.

8. Substitute the values of n back into the expression for the consecutive integers:
If n = -2: -2, -1, 0
If n = 1: 1, 2, 3

Therefore, the three sets of consecutive integers that satisfy the given condition are:
-2, -1, 0 and 1, 2, 3