Jason is designing a rectangular garden that has area of 300 square feet. The width of the garden is five feet less than its length. Find its length and width
To find the length and width of the rectangular garden, we can set up an equation based on the given information.
Let's say the length of the garden is represented by L, and the width is represented by W.
According to the given information, the area of the garden is 300 square feet. The formula for the area of a rectangle is length times width, so we can set up the equation:
L * W = 300
Additionally, it is stated that the width of the garden is five feet less than its length. Mathematically, this can be written as:
W = L - 5
Now, we can substitute this expression for W in terms of L into the equation for the area:
L * (L - 5) = 300
Expanding this equation, we get:
L^2 - 5L = 300
Rearranging the equation to bring all terms to one side, we have:
L^2 - 5L - 300 = 0
Now, we have a quadratic equation. To solve for L, we can factor it or use the quadratic formula.
Factoring the above equation, we find:
(L - 20)(L + 15) = 0
From this equation, we can see that either L - 20 = 0 or L + 15 = 0.
If L - 20 = 0, solving for L gives us L = 20.
If L + 15 = 0, solving for L gives us L = -15. However, since the length of a garden cannot be negative, we discard this solution.
Therefore, the length of the rectangular garden is 20 feet.
To find the width, we substitute the value of L into the equation W = L - 5:
W = 20 - 5
Simplifying, we find W = 15.
Therefore, the length of the garden is 20 feet, and the width is 15 feet.
Let's solve this step-by-step:
Step 1: Assign variables
Let's assign variables to the length and width of the garden. Let L be the length (in feet) and W be the width (in feet).
Step 2: Write the given information as equations
We are given two pieces of information:
1) The area of the garden is 300 square feet.
2) The width is five feet less than the length.
We can write these as equations:
1) L * W = 300
2) W = L - 5
Step 3: Solve the equations
We have two equations:
1) L * W = 300
2) W = L - 5
Substitute equation 2) into equation 1):
L * (L - 5) = 300
Expand and simplify:
L^2 - 5L = 300
Rearrange the equation:
L^2 - 5L - 300 = 0
Step 4: Factor or use the quadratic formula to solve for L
The equation can be factored as follows:
(L - 20)(L + 15) = 0
Setting each factor equal to zero:
L - 20 = 0 or L + 15 = 0
Solving for L, we have two possible solutions:
L = 20 or L = -15
Since the length cannot be negative, we discard L = -15 as an extraneous solution.
So, the length of the garden is L = 20 feet.
Step 5: Calculate the width
Using equation 2) W = L - 5
W = 20 - 5
W = 15
Therefore, the width of the garden is W = 15 feet.
In conclusion, the length of the garden is 20 feet and the width is 15 feet.