A rectangle pen for a pet is 5 feet longer than it is wide. Give possible values for the width W of the pen if its area must be between 266 and 750 square feet, inclusively. The width of the pen from the smaller value of the width is how many feet to the larger value of the width, and how many feet inclusively
To solve this problem, we can start by setting up an equation based on the given information.
Let's assume the width of the rectangle pen is W feet. According to the problem, the length of the pen is 5 feet longer than the width, so the length would be (W + 5) feet.
The area of a rectangle is calculated by multiplying its length by its width. Therefore, the area of the pen can be expressed as:
Area = Width * Length
Area = W * (W + 5)
Now, we are given that the area of the pen must be between 266 and 750 square feet, inclusively. So, we can set up an inequality to represent this condition:
266 ≤ W * (W + 5) ≤ 750
To solve this inequality, we need to find the possible values of W that satisfy it. Let's solve it step by step:
1. Lower Bound:
266 ≤ W * (W + 5)
Expand the expression:
266 ≤ W^2 + 5W
Rearrange the equation to bring all terms to one side:
W^2 + 5W - 266 ≥ 0
Now, use factoring or the quadratic formula to solve the equation:
(W - 14)(W + 19) ≥ 0
The possible values for W that satisfy this inequality are W ≤ -19 or W ≥ 14.
2. Upper Bound:
W * (W + 5) ≤ 750
Expand the expression:
W^2 + 5W ≤ 750
Rearrange the equation to bring all terms to one side:
W^2 + 5W - 750 ≤ 0
Once again, factor or use the quadratic formula to solve the equation:
(W - 25)(W + 30) ≤ 0
The possible values for W that satisfy this inequality are -30 ≤ W ≤ 25.
Combining both bounds, we find that the possible values for the width W of the rectangle pen are:
-30 ≤ W ≤ -19, and 14 ≤ W ≤ 25.
Therefore, the width of the pen ranges from -30 feet to -19 feet of the smaller value, and from 14 feet to 25 feet of the larger value inclusively.