A rancher has 200 feet of fencing to enclose two adjacent corrals
1 answer:
Answer:
a) Each corral should be 33⅓ ft long and 25 ft wide
b) The total enclosed area is 1666⅔ ft²
Step-by-step explanation:
I assume that the corrals have identical dimensions and are to be fenced as in the diagram below
Let x = one dimension of a corral
and y = the other dimension
(a) Dimensions to maximize the area
The total length of fencing used is:
4x + 3y = 200
4x = 200 – 3y
x = 50 - ¾y
The area of one corral is A = xy, so the area of the two corrals is
A = 2xy
Substitute the value of x
A = 2(50 - ¾y)y
A = 100 y – (³/₂)y²
This is the equation for a downward-pointing parabola:
A = (-³/₂)y² + 100y
a = -³/₂; b = 100; c = 0
The vertex (maximum) occurs at
y = -b/(2a) = 100 ÷ (2׳/₂) = 100 ÷ 3 = 33⅓ ft
4x + 3y = 100
Substitute the value of y
4x + 3(33⅓) = 200
4x + 100 = 200
4x = 100
x = 25 ft
Each corral should measure 33⅓ ft long and 25 ft wide.
Step 2. Calculate the total enclosed area
The enclosed area is 50 ft long and 33⅓ ft wide.
A = lw = 50 × 100/3 = 5000/3 = 1666⅔ ft²
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Answer:
j=13, g=20.8, h=24
Step-by-step explanation:
The overall shape given and the shape within, are both right triangles. With right triangles, you are allowed to use the pythagorean theorem formula () in order to solve for some sides. In this case, that would be j and h. The five in the smaller triangle is represented by b and the 12 is the hypotenuse so it is represented by c. When you plug in those numbers in the pythagorean theorem formula, you will find the value of j to be 13. When looking at this, we see that 12 is the second greatest value in the right triangle values that we just found, so we know the the opposing angle for that one will be 60 degrees. The 5's opposing side is therefore 30 degrees. When subtracting 90 and 30, we get 60, so therefore you can use the 30 60 90 formula to find the sides of the bigger triangle. The 60 degrees represents g. This formula will be
. The a is 12 since it is the smallest value. So therefore, g is
, which is 20.8. Now that we have this side, we can just use the pythagorean theorem formula to find the remaining side. Therefore, h is going to be 24