Geometry/Maximum area: A rancher wants to enlarge an existing rectangular corral such that the total area of the new corral is 1.5 times that of the original corral. The current corral's dimensions are 250 feet by 160 feet. The rancher wants to increase each dimension by the same amount.
1.) write a function that represents the area, A, of the new corral.
2.) Find the dimensions of the new corral.
250*160 = ____
A = Length * Width * Expansion Factor
A = 1.5*250*160
present area = 250(160) = 40000
new area = 1.5(40000) = 60000
let the new length be 250+x
let the new width be 160+x
(250+x)(160+x) = 60000
solve for x,
Hint, use the quadratic equation, since it does not factor.
1.) To write a function that represents the area, A, of the new corral, we can start by defining the original length and width as L and W respectively, and let x represent the amount by which each dimension will be increased.
The length of the new corral will be L + x, and the width will be W + x.
The area, A, of the new corral can be represented by the product of the length and width:
A = (L + x) * (W + x)
So the function that represents the area, A, of the new corral is:
A(x) = (L + x) * (W + x)
2.) To find the dimensions of the new corral, we need to solve for x in the equation A(x) = 1.5 * A, where A is the area of the original corral.
Substituting the known values, we have:
A(x) = (250 + x) * (160 + x)
1.5 * A = 1.5 * (250 * 160)
Expanding the equation, we get:
(250 + x) * (160 + x) = 1.5 * 250 * 160
Simplifying further, we have:
40000 + 410x + x^2 = 60000
Rearranging the equation to solve for x:
x^2 + 410x + 40000 - 60000 = 0
x^2 + 410x - 20000 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 410, and c = -20000. Plugging in the values, we have:
x = (-410 ± √(410^2 - 4(1)(-20000))) / (2(1))
Simplifying, we get:
x = (-410 ± √(168100 + 80000)) / 2
x = (-410 ± √(248100)) / 2
x ≈ (-410 ± 498.1) / 2
This gives us two possible solutions for x:
1) x ≈ (-410 + 498.1) / 2 ≈ 44.05
2) x ≈ (-410 - 498.1) / 2 ≈ -454.05
Since we are looking for positive lengths for the dimensions of the new corral, we can disregard the second solution.
Therefore, the dimensions of the new corral will be approximately 250 + 44.05 feet by 160 + 44.05 feet.
To solve this problem, we can follow these steps:
1.) Write a function that represents the area, A, of the new corral.
Let's denote the increase in dimensions by 'x'. Then the new dimensions of the corral would be (250 + x) feet by (160 + x) feet. The area of the new corral can be calculated by multiplying these dimensions:
A = (250 + x)(160 + x)
This equation represents the area, A, of the new corral as a function of the increase in dimensions, x.
2.) Find the dimensions of the new corral.
To find the dimensions of the new corral, we need to solve the equation for x. We know that the total area of the new corral is 1.5 times the area of the original corral, which is 250 * 160.
1.5 * (250 * 160) = (250 + x)(160 + x)
Now, we can solve this equation to determine the value of x.
(1.5 * 250 * 160) = (250 + x)(160 + x)
(1.5 * 250 * 160) = 40000 + 410x + x^2
We can rearrange the equation to get it in quadratic form:
0 = x^2 + 410x + (40000 - (1.5 * 250 * 160))
Now, we can solve this quadratic equation to find the value of x using factoring, completing the square, or the quadratic formula.