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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.

 Apr 30, 2024
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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.

 Apr 30, 2024

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