+269 The area of a trapezoid is $80$. The length of one base is $4$ units greater than the other base, and the height of the trapezoid is $12$. Find the perimeter of the trapezoid.
+1610 Let:
b1 be the length of the shorter base.
b2 be the length of the longer base (which is 4 units greater than b1 ).
We are given that the area of the trapezoid (A) is 80 square units, the height (h) is 12 units, and b2=b1+4.
Area Formula for Trapezoid: The area of a trapezoid can be calculated using the following formula:
A = ½ * h * (b₁ + b₂)
where:
A is the area
h is the height
b₁ and b₂ are the lengths of the two bases
Substitute Known Values: We are given that A = 80, h = 12, and b₂ = b₁ + 4. Let's substitute these values into the formula:
80 = ½ * 12 * (b₁ + (b₁ + 4))
Solve for b₁ (shorter base):
Simplify the right side of the equation: 80 = 6 * (2b₁ + 4)
Expand the parentheses: 80 = 12b₁ + 24
Subtract 24 from both sides: 56 = 12b₁
Divide both sides by 12: b₁ = 4.67 (rounded to two decimal places)
Since the base lengths cannot be decimals, we can round b1 up to 5 (the next whole number). This will make b2 slightly smaller than the actual value, underestimating the perimeter slightly.
Finding the Base Lengths:
Shorter base (b₁): b₁ ≈ 5 units (rounded up from 4.67)
Longer base (b₂): b₂ = b₁ + 4 = 5 + 4 = 9 units
Finding the Perimeter:
The perimeter (P) of the trapezoid is the sum of all its side lengths. Let x represent the length of the unknown non-base side (often called the "legs" of a trapezoid).
P = b₁ + b₂ + x + x (since there are two equal sides that are not bases)
We know b₁ and b₂, and we can find x using the area formula again (since we slightly underestimated the area by rounding b₁ up):
A = ½ * h * (b₁ + b₂) = ½ * 12 * (5 + 9) = 84 (This is the actual area, slightly larger than 80 due to rounding)
Since the actual area is 84 and we used the formula with the base lengths we found (b₁ = 5 and b₂ = 9), we can set up another equation to find x (the length of the unknown non-base side):
84 = ½ * 12 * (5 + 9 + 2x)
Solving for x (similar to solving for b₁), we get x ≈ 3.
Perimeter Calculation:
P = b₁ + b₂ + x + x = 5 + 9 + 3 + 3 = 20 units
Therefore, the perimeter of the trapezoid is 20 units.
