The perimeter of a rectangle is 200 m, and its area is 1875 m^2. Find its dimensions.
i did:
2x+2y=200
2x/2=200/2-2y/2
x=100 -y
(100-y)*y = 1875
y^2-100+1875=0
y^2 - 25y-75y +1875=0
y(y-25)-75(y-25)=0
(y-25)(y-75)=0
length= 75
width = 25
good job,
small typo in
y^2-100+1875=0
should be
y^2-100y+1875=0
steps are ok, answer is correct
i must have skipped it when i typing that y but thank you.
Your calculations and solution are correct.
From your equation 2x + 2y = 200, you substituted x = 100 - y. This gives you the equation (100 - y) * y = 1875.
Simplifying further, you have y^2 - 100y + 1875 = 0.
Factoring this quadratic equation, you get (y - 25)(y - 75) = 0.
So, y = 25 or y = 75.
If y = 25, then x = 100 - y = 100 - 25 = 75. This gives you the dimensions 75 m (length) and 25 m (width).
If y = 75, then x = 100 - y = 100 - 75 = 25. This also gives you the dimensions 25 m (length) and 75 m (width).
Therefore, the dimensions of the rectangle can be either 75 m by 25 m or 25 m by 75 m.
To find the dimensions of the rectangle given its perimeter and area, you correctly set up the equations:
1. Perimeter equation: 2x + 2y = 200
2. Area equation: x * y = 1875
Now, let's walk through the steps to solve these equations:
1. Perimeter equation (2x + 2y = 200):
- Simplifying: Divide both sides of the equation by 2: x + y = 100.
- Rearranging the equation: x = 100 - y.
2. Area equation (x * y = 1875):
- Substituting the value of x from the perimeter equation: (100 - y) * y = 1875.
- Expanding: y^2 - 100y + 1875 = 0.
From here, you applied the quadratic formula and factored correctly:
3. Solving the quadratic equation (y^2 - 100y + 1875 = 0):
- Factoring: (y - 25)(y - 75) = 0.
- Setting each factor equal to zero: y - 25 = 0 or y - 75 = 0.
- Solving for y: y = 25 or y = 75.
Now that you have two possible values for y, you need to calculate the corresponding values for x:
4. Calculating the dimensions (length and width):
- If y = 25, substitute y = 25 into the perimeter equation: x + 25 = 100 => x = 75.
- If y = 75, substitute y = 75 into the perimeter equation: x + 75 = 100 => x = 25.
Hence, the dimensions of the rectangle are:
- Length = 75 m and Width = 25 m, or
- Length = 25 m and Width = 75 m.