Questions: In the National Hockey League, the goalie may not play the puck outside the isosceles trapezoid behind the net. The formula for the area of a trapezoid is A=(1/2)(b1+b2)h. Complete parts (a)-(c). a. Solve the formula for the base b1. b1=(2A/h)-b2 b. Use the formula to find the length of the base next to the goal given that the height of the trapezoid is 11 ft and the base farthest from the goal is 28 ft.

In the National Hockey League, the goalie may not play the puck outside the isosceles trapezoid behind the net. The formula for the area of a trapezoid is A=(1/2)(b1+b2)h. Complete parts (a)-(c).

a. Solve the formula for the base b1.
b1=(2A/h)-b2

b. Use the formula to find the length of the base next to the goal given that the height of the trapezoid is 11 ft and the base farthest from the goal is 28 ft.

Transcript text: In the National Hockey League, the goalie may not play the puck outside the isosceles trapezoid behind the net. The formula for the area of a trapezjsid is $\mathrm{A}=\frac{1}{2}\left(b_{1}+b_{2}\right) \mathrm{h}$. Complete parts (a)-(c). a. Solve the formula for the base $b_{1}$. \[ b_{1}=\frac{2 A}{h}-b_{2} \] b. Use the formula to find the length of the base next to the goal given that the height of the trapezoid is 11 ft and the base farthest from the goal is 28 ft.

Solution

Solution Steps

Solution Approach

a. To solve the formula for the base $ b_1 $, we need to isolate $ b_1 $ on one side of the equation. Start with the formula for the area of a trapezoid: $ A = \frac{1}{2}(b_1 + b_2)h $. Multiply both sides by 2 to eliminate the fraction, then solve for $ b_1 $.

b. Use the formula derived in part (a) to find the length of the base next to the goal. Substitute the given values for the height $ h = 11 $ ft, the area $ A $, and the base farthest from the goal $ b_2 = 28 $ ft into the formula to calculate $ b_1 $.

Step 1: Solve the Formula for $ b_1 $

To solve the formula for the base $ b_1 $, start with the area of a trapezoid formula: \[ A = \frac{1}{2}(b_1 + b_2)h \] Multiply both sides by 2 to eliminate the fraction: \[ 2A = (b_1 + b_2)h \] Divide both sides by $ h $ to isolate $ b_1 + b_2 $: \[ b_1 + b_2 = \frac{2A}{h} \] Subtract $ b_2 $ from both sides to solve for $ b_1 $: \[ b_1 = \frac{2A}{h} - b_2 \]

Step 2: Calculate $ b_1 $ Given $ h = 11 $ ft, $ b_2 = 28 $ ft, and $ A = 100 $

Substitute the given values into the formula: \[ b_1 = \frac{2 \times 100}{11} - 28 \] Calculate the expression: \[ b_1 = \frac{200}{11} - 28 \approx 18.1818 - 28 \] \[ b_1 \approx -9.8182 \]