The width of a rectangle is 1/3 as long as its length. If the difference between the perimeter of the rectangle and its length is 60 cm, find the area of the rectangle.
2(w + 3w) - 3w = 60
w = 12
so the area is 12 * 36 = 432
Let's break down the problem step by step:
Step 1: Let's assume the length of the rectangle is L cm.
Step 2: According to the problem, the width of the rectangle is 1/3 as long as its length, so the width is (1/3)L cm.
Step 3: The formula for the perimeter of a rectangle is P = 2(L + W), where P is the perimeter, L is the length, and W is the width. So, the perimeter of the rectangle is P = 2(L + (1/3)L) = 2(4/3)L = (8/3)L cm.
Step 4: According to the problem, the difference between the perimeter and the length of the rectangle is 60 cm. So, (8/3)L - L = 60 cm.
Step 5: Simplifying the equation from step 4, we get:
(8/3)L - (3/3)L = 60 cm,
(5/3)L = 60 cm,
L = (60 cm) * (3/5) = 36 cm.
Step 6: Now that we know the length of the rectangle is 36 cm, we can find the width by multiplying the length by (1/3):
W = (1/3)L = (1/3)(36 cm) = 12 cm.
Step 7: The area of a rectangle is given by the formula A = L * W. So, the area of this rectangle is:
A = (36 cm) * (12 cm) = 432 cm^2.
Therefore, the area of the rectangle is 432 cm^2.
To find the area of the rectangle, we need to determine its length and width.
Let's assume that the length of the rectangle is "L" cm. According to the problem, the width is 1/3 as long as the length. Therefore, the width can be expressed as (1/3) × L = L/3 cm.
The formula for the perimeter of a rectangle is given by: P = 2(L + W), where P is the perimeter, L is the length, and W is the width.
Based on the problem statement, we know that the difference between the perimeter of the rectangle and its length is 60 cm. So the equation can be written as: P - L = 60 cm.
Substituting the values of the perimeter and the lengths, we get: 2(L + (L/3)) - L = 60.
Simplifying the equation, we can find the value of L:
2(4L/3) - L = 60
(8L/3) - L = 60
8L - 3L = 180 (multiplying both sides by 3 to eliminate the fraction)
5L = 180
L = 36 cm
Now that we have found the length of the rectangle (L = 36 cm), we can calculate the width using the formula W = L/3:
W = 36/3 = 12 cm
Finally, we can find the area of the rectangle using the formula A = L × W:
A = 36 cm × 12 cm
A = 432 cm²
Therefore, the area of the rectangle is 432 square centimeters.