The width of a rectangle is three fourths the length. The perimeter of the rectangle becomes 50 cm when the length and the width are each increased by 2cm. Find the length and the width.
Draw a diagram and convince yourself that the perimeter P=2(L+W)
We're told that W=(3/4)L.
We're also told that P=50 when W+2 and L+2 are used. So
50=2((L+2) + (W+2))=2((L+2) + ((3/4)L + 2))
simplifying
25=(L + (3/4)L + 4)
21=(7/4)L
Solve for L then use the relation between W and L given above, i.e. the first sentence of the problem.
To find the length and width of the rectangle, we can start by drawing a diagram to visualize the problem. Let's assign the length of the rectangle as L and the width as W.
From the given information, we know that the width is three-fourths the length, so we can write the equation:
W = (3/4)L
The perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, the formula for the perimeter (P) is:
P = 2(L + W)
We're also given that when both the length and width are increased by 2 cm, the new perimeter is 50 cm. So we can write another equation:
50 = 2((L + 2) + (W + 2))
Simplifying the second equation, we have:
50 = 2(L + 2) + 2W + 4
50 = 2L + 4 + 2W + 4
50 = 2L + 2W + 8
Now we can substitute the value of W from the first equation into the second equation:
50 = 2L + 2((3/4)L) + 8
50 = 2L + (3/2)L + 8
50 = (4/2)L + (3/2)L + 8
50 = (7/2)L + 8
Next, we can simplify the equation by subtracting 8 from both sides:
50 - 8 = (7/2)L + 8 - 8
42 = (7/2)L
To solve for L, we can multiply both sides by (2/7):
L = (42 * 2)/7
L = 12 cm
Now that we have the length, we can substitute it back into the first equation to find the width:
W = (3/4)L
W = (3/4)(12)
W = 9 cm
Therefore, the length of the rectangle is 12 cm and the width is 9 cm.