Mr. smith wants to buy some fencing to completely surround his rectangular garden. He knows that the total perimeter of the garden is 26 meters and the total area is 40 square meters. What is the garden’s length and width?
Possible dimensions:
2 by 20
4 by 10
5 by 8
Which of those dimensions will give a perimeter of 26 meters?
To find the garden's length and width, we can set up a system of equations based on the given information. Let's assume that the length of the garden is "L" and the width is "W".
1. Perimeter equation: The total perimeter of a rectangle is equal to the sum of all sides. In this case, we have two sides of length "L" and two sides of length "W". The equation for the perimeter is: 2L + 2W = 26.
2. Area equation: The area of a rectangle is equal to its length multiplied by its width. In this case, the area is given as 40 square meters. The equation for the area is: L * W = 40.
Now, we have a system of equations:
Equation 1: 2L + 2W = 26
Equation 2: L * W = 40
We can solve this system of equations to find the values of "L" and "W".
Let's use the substitution method to solve the system:
Step 1: Solve Equation 2 for L (Express L in terms of W):
L = 40 / W
Step 2: Substitute the value of L from Step 1 into Equation 1:
2(40 / W) + 2W = 26
Step 3: Simplify the equation:
80 / W + 2W = 26
Step 4: Multiply the entire equation by W to eliminate the fraction:
80 + 2W^2 = 26W
Step 5: Rearrange the equation to a quadratic equation form:
2W^2 - 26W + 80 = 0
Step 6: Solve the quadratic equation. Factorization or quadratic formula can be used.
By using the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / 2a
In our case:
a = 2, b = -26, c = 80
W = (-(-26) ± √((-26)^2 - 4(2)(80))) / 2(2)
W = (26 ± √(676 - 640)) / 4
W = (26 ± √36) / 4
W = (26 ± 6) / 4
This gives us two possible values for W:
W1 = (26 + 6) / 4 = 8
W2 = (26 - 6) / 4 = 5
Step 7: Substitute the values of W into Equation 2 to find the corresponding values of L.
For W = 8: L = 40 / 8 = 5
For W = 5: L = 40 / 5 = 8
Therefore, the width of the garden can be either 5 meters and the length 8 meters, or the width can be 8 meters and the length 5 meters.