Marcus has 68 ftof fencing. He wants to build a rectangular pen with the largest possible area. What should the dimensions of the rectangle be?
a. 19 ft by 21 ft
b. 21 ft by 13 ft
c. 17 ft by 17 ft
d. 19 ft by 15 ft
I got a because I multiplied 19 x 21 and I got 399. Is this right?
First, you have to check to make sure you have sufficient fencing. Take each of the four options and calculate the perimeter to make sure you have no more than 68 ft.
For the first one, the perimeter would be 19+19+21+21 or 19(2) + 21(2), both of which equal 80. You don't have enough fencing for that, so eliminate the option.
Then, with your remaining options, calculate the area by multiplying the sides as you did above to see which is largest.
Is it c. 17 by 17?
Yes, that's right.
17 + 17 + 17 + 17 = 68
or
17(4) = 68
To solve this problem, we can use the concept of maximizing the area of a rectangle given a fixed perimeter. In this case, the fixed perimeter is 68 ft of fencing.
Let's consider the formula for the perimeter of a rectangle:
Perimeter = 2(length + width)
Knowing that the perimeter is 68 ft, we can rewrite the formula as:
68 = 2(length + width)
Now, let's express one variable in terms of the other to eliminate one variable. Let's solve for length in terms of width:
length = (68 - 2width) / 2
length = 34 - width
We want to maximize the area of the rectangle, which is given by the formula:
Area = length x width
Substituting the expression for length, we get:
Area = (34 - width)x(width)
Area = 34w - w²
To maximize the area, we need to find the value of width that gives us the maximum value for the area. One way to do this is by using calculus, but we can also approximate it by considering the given options.
Now, let's calculate the areas for each option and see which one is the largest:
Option a: 19 ft by 21 ft
Area = 19 x 21 = 399 sq ft
Option b: 21 ft by 13 ft
Area = 21 x 13 = 273 sq ft
Option c: 17 ft by 17 ft
Area = 17 x 17 = 289 sq ft
Option d: 19 ft by 15 ft
Area = 19 x 15 = 285 sq ft
Based on the calculations, it seems that option a has the largest area. However, to be certain, let's compare all the areas again:
Option a: 399 sq ft
Option b: 273 sq ft
Option c: 289 sq ft
Option d: 285 sq ft
Therefore, the correct answer is option a, 19 ft by 21 ft, which gives you an area of 399 sq ft.
Your calculation of 19 x 21 as 399 sq ft is correct, and it matches the answer we derived using the given conditions. Well done!