use complecompleteng the squares to find the dimension of the rectangular board
Answers
Step-by-step explanation:
Ok, let's remember Area and Perimeter formulas for rectangle:
A = L*W
P = 2L+2W
We are given a relationship between the length and width of the rectangle. The length is 8 cm longer than the width. Write this as an equation.
L = W+8
We are given the area of the rectangle, so let's plug this into the area formula.
250 = L*W
Now, to solve for the width, let's look at the 3 equations we have now:
1) 250 = L*W (Area)
2) L = W+8
3) P = 2L+2W
The 3rd equation involves perimeter, and we're not there yet, so just focus on equations 1 & 2.
Notice that we have 2 equations with 2 variables (L and W). We can solve by substitution.
Plugging in (W+8) in place of L:
250 = (W+8)*W
Solve for W:
250 = W2+8W
Now we have a quadratic equation (0 = W2+8W - 250), so we cannot simply solve algebraically.
We need to complete the square.
Step 1) Divide all terms by the leading coefficient (coefficient of W2), which is 1, so no change.
Step 2) Move constant term to other side of equation by itself.
250 = W2+8W
Step 3) Find the number needed to complete the square by dividing the middle term coefficient (8) in half, and then squaring it. 8/2=4. 42=16
Step 4) Complete the square by adding this number (16) to both sides of the equation.
250+16=W2+8W+16
Now we can factor the RHS of the equation. It will be a perfect square.
266=(W+4)^2
Now we can use a calculator to take the square root of 266.
16.3095=W+4
W = 12.3095 cm
We found the width of the rectangle! Don't forget units were given as cm.
Now, plug this into the equation with L to solve for the length:
L = W+8
L = (12.3095)+8
L = 20.3095 cm
Double check by multiplying L*W = Area.
20.3095 * 12.3095 = 250
Perfect!
Now find the perimeter:
P = 2L+2W
P = 2(20.3095) + 2(12.3095)
P = 65.238 cm
Remember to write the units, cm.
That's the answer! Hope that helps! :)