Question

A trapezoid has an area of 20 cm2 and a height z cm. The lengths of the parallel sides are (2z + 3) cm and (6z – 1) cm. Find the height, z, of the trapezoid. In your final answer, include all of the formulas and calculations necessary.

Answer 1

To find the height z of a trapezoid with given parallel side lengths, apply the area formula of a trapezoid and solve the resulting quadratic equation to determine the appropriate value of z.

Step-by-step explanation:

To find the height z of the trapezoid with an area of 20 cm² and parallel sides of (2z + 3) cm and (6z – 1) cm, we use the area formula of a trapezoid which is Area = 0.5 × (sum of the parallel sides) × height. Substituting our known values into the formula gives:

20 = 0.5 × ((2z + 3) + (6z – 1)) × z.

First, we simplify the sum of the parallel sides:

2z + 3 + 6z – 1 = 8z + 2.

Substitute back into the area equation and solve for z:

20 = 0.5 × (8z + 2) × z

40 = (8z + 2) × z

40 = 8z² + 2z

This quadratic equation can then be solved using factorization or the quadratic formula. After finding solutions for z, we must choose the value that makes sense in the context of the trapezoid's dimensions (i.e., a positive height).

Answer 2

Area of a trapezoid = ½(a + b)h, where a and b are the lengths of the parallel sides and h is its height.

From your information 20 = ½(2z + 3 + 6z – 1)z = ½(8z + 2)z = z(4z + 1)

Solve 20 = 4z² + z which is 0 = 4z² + z – 20 using the quadratic formula