Math

QuestionA farmer has 300 feet of fence for a 4500 sq ft area of adjoining squares with sides xx and yy. Find xx and yy. x= x= y= y=

Studdy Solution

STEP 1

Assumptions1. The total length of the fence available is300 feet. . The total area to be enclosed is4500 square feet.
3. The region is in the shape of two adjoining squares, one larger with side length x and one smaller with side length y.
4. The sides of the squares are the only parts that require fencing.

STEP 2

First, we need to set up the equation for the total length of the fence, which is the sum of the perimeters of the two squares.Perimeter=x+4yPerimeter =x +4y

STEP 3

Next, we set up the equation for the total area of the two squares, which is the sum of the areas of the two squares.
Area=x2+y2Area = x^2 + y^2

STEP 4

We know the total length of the fence is300 feet and the total area is4500 square feet. So, we can substitute these values into our equations.
3x+4y=3003x +4y =300x2+y2=4500x^2 + y^2 =4500

STEP 5

We can solve the first equation for y to get an equation in terms of x.
y=(3003x)/4y = (300 -3x) /4

STEP 6

Substitute the equation from5 into the second equation from4.
(x2)+((3003x)/4)2=4500(x^2) + ((300 -3x) /4)^2 =4500

STEP 7

olve the equation from6 for x. This will give us a quadratic equation.
16x224x+900=7200016x^2 -24x +900 =72000

STEP 8

Subtract72000 from both sides of the equation to set it equal to zero.
16x224x63000=016x^2 -24x -63000 =0

STEP 9

Divide the entire equation by16 to simplify it.
x2.5x3937.5=x^2 -.5x -3937.5 =

STEP 10

olve the quadratic equation from9 for x using the quadratic formula.
x=[.5±sqrt((.5)24(3937.5))]/2x = [.5 ± sqrt((.5)^2 -4(-3937.5))]/2

STEP 11

Calculate the two possible values for x.
x=[.5±sqrt(.25+15750)]/x = [.5 ± sqrt(.25 +15750)]/

STEP 12

implify the equation to find the two possible values for x.
x=[.5±sqrt(15752.25)]/2x = [.5 ± sqrt(15752.25)]/2

STEP 13

Calculate the two possible values for x.
x=[.5±125.5]/2x = [.5 ±125.5]/2

STEP 14

Calculate the two possible values for x.
x=63.,62x =63., -62

STEP 15

Since x represents a length, it cannot be negative. So, we discard the negative solution.
x=63.5x =63.5

STEP 16

Substitute x =63.5 into the equation from5 to find the value of y.
y=(300363.5)/4y = (300 -3*63.5) /4

STEP 17

Calculate the value of y.
y=36.5y =36.5The lengths of the sides of the squares are x =63.5 feet and y =36.5 feet.

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