Final answer:
To find the different sets of dimensions for Dana's goat pen, we must satisfy the minimum length requirement and the maximum perimeter requirement. We can use the formula for the perimeter of a rectangle and rearrange it to solve for the width. By substituting the values and setting up an inequality, we can determine the range of possible lengths and calculate the number of different sets of dimensions.
Step-by-step explanation:
To find the different sets of dimensions for Dana's goat pen, we need to consider the requirements given. The length of the pen should be at least 50 feet, and the perimeter should be no more than 190 feet. Let's denote the length as L and the width as W.
The perimeter of a rectangle is given by the formula P = 2(L + W). We can rearrange this formula to solve for W as:
W = (P - 2L) / 2
Substituting the given values, we have:
W = (190 - 2L) / 2
To ensure that the length is at least 50 feet, we can set up the following inequality:
L ≥ 50
Now, let's substitute the value of W we found into the inequality:
L ≥ 50
(190 - 2L) / 2 ≥ 50
190 - 2L ≥ 100
-2L ≥ -90
L ≤ 45
So, the values of L can range from 50 to 45 feet, in increments of 1 foot. This gives us a total of 6 different sets of dimensions for the goat pen.