The manager at a daycare facility wants to put a fence around a rectangular play yard just outside of the building. she will use a wall of the building as one side of this rectangle, so that fencing is only needed on the other three sides. if there are 96 feet of fencing available, and x represents the length of one of the sides of the rectangle that is perpendicular to the building's wall, then find the value of x that will maximize the area of this play yard. (see the picture below)

24

Step-by-step explanation:

If x is the length of the perpendicular sides, then the length of the parallel side must by 96−2x.

The total area is:

A = x (96 − 2x)

A = 96x − 2x²

This is a parabola, so we know the maximum is at the vertex (at x = -b/(2a)).

x = -96 / (2 · -2)

x = 24

Or, we can optimize using calculus.  Find dA/dx and set to 0:

dA/dx = 96 − 4x

0 = 96 − 4x

x = 24

The value of x that maximizes the area is x = 24 ft.

x ≥ 18 is your answer

step-by-step explanation:

let "a number" = x

(2/3)x + 17 ≥ 29

isolate the variable (x). do the opposite of pemdas (parenthesis, exponent (& root), multiplication, division, addition, subtraction).

first, subtract 17 from both sides.

(2/3)x + 17 (-17) ≥ 29 (-17)

(2/3)x ≥ 29 - 17

(2/3)x ≥ 12

next, multiply 3/2 to both sides to isolate the x

(3/2)(2/3)x ≥ 12(3/2)

x ≥ 12(3/2)

simplify. multiply, then divide

x ≥ (12 * 3)/2

x ≥ (36)/2

x ≥ 18

x ≥ 18 is your answer

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