Math

QuestionA farmer has 250 ft of fence for two adjoining squares with area 3125 sq ft. Find side lengths xx and yy.

Studdy Solution

STEP 1

Assumptions1. The total length of the fence available is250 feet. . The total area to be enclosed is3125 square feet.
3. The region to be enclosed is in the shape of two adjoining squares.
4. The larger square has sides of length xx.
5. The smaller square has sides of length yy.
6. The squares share a full side.

STEP 2

First, we need to set up an equation for the total length of the fence. The total length of the fence is the sum of the perimeters of the two squares, minus the length of the side they share.
Totallength=4x+4yyTotal\, length =4x +4y - y

STEP 3

Now, plug in the total length of the fence available to set up the equation.
250=x+yy250 =x +y - y

STEP 4

implify the equation.
250=4x+3y250 =4x +3y

STEP 5

Next, we need to set up an equation for the total area to be enclosed. The total area is the sum of the areas of the two squares.
Totalarea=x2+y2Total\, area = x^2 + y^2

STEP 6

Now, plug in the total area to be enclosed to set up the equation.
3125=x2+y23125 = x^2 + y^2

STEP 7

Now we have a system of two equations, which are4x+3y=2504x +3y =250x2+y2=3125x^2 + y^2 =3125

STEP 8

We can solve this system of equations by substitution or elimination. Let's use substitution. First, solve the first equation for xx.
x=(2503y)/4x = (250 -3y) /4

STEP 9

Substitute xx in the second equation.
(2503y)2/16+y2=3125(250 -3y)^2 /16 + y^2 =3125

STEP 10

implify the equation.
1562518750y+9y2+16y2=5000015625 -18750y +9y^2 +16y^2 =50000

STEP 11

Combine like terms.
25y18750y+15625=5000025y^ -18750y +15625 =50000

STEP 12

Subtract50000 from both sides to set the equation to zero.
25y218750y34375=025y^2 -18750y -34375 =0

STEP 13

Divide the equation by25 to simplify.
y2750y1375=0y^2 -750y -1375 =0

STEP 14

olve this quadratic equation for yy using the quadratic formula.
y=[750±sqrt((750)24(137))]/(2)y = [750 ± sqrt((750)^2 -4**(-137))] / (2*)

STEP 15

Calculate the discriminant.
Discriminant=(750)24(1375)=562500+5500=568000Discriminant = (750)^2 -4**(-1375) =562500 +5500 =568000

STEP 16

Substitute the discriminant back into the quadratic formula to find the values of yy.
y=[750±sqrt(568000)]/2y = [750 ± sqrt(568000)] /2

STEP 17

Calculate the two possible values for yy.
y=[750±754]/2y = [750 ±754] /2

STEP 18

Calculate the two possible values for yy.
y=2,752y = -2,752

STEP 19

Since yy cannot be negative (as it represents a length), we discard - and take y=752y =752.

STEP 20

Substitute y=752y =752 into the first equation to find xx.
4x+3752=2504x +3*752 =250

STEP 21

implify the equation.
4x+2256=2504x +2256 =250

STEP 22

Subtract2256 from both sides to isolate xx.
4x=25022564x =250 -2256

STEP 23

Calculate the value of xx.
x=2006x = -2006

STEP 24

Divide both sides by4 to solve for xx.
x=2006/4x = -2006 /4

STEP 25

Calculate the value of xx.
x=501.5x = -501.5

STEP 26

Since xx cannot be negative (as it represents a length), we discard 501.5-501.5 and take x=0x =0.
The lengths of the sides of the two squares are x=0x =0 and y=752y =752.

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