Two streets bounding your triangular lot make an angle of 74∘. The lengths of the two sides of the lot on these streets are 126 feet and 110 feet. You want to build a fence on the third side, but have only 150 feet of fencing on hand. a. Do you have enough fencing? Justify your answer. b. What are the measures of the other two angles of the lot? c. The city has zoned the property so that any residence must have a square footage at least one-third the area of the lot itself. You plan to build a 2300ft2 home. Will the city approve your plans? Why or why not?

Answer: a) Yes, there is enough fance

b) 58.1° and 47.9°

c) The city will not approve, because 1/3 of the area is just 2220.5ft²

Step-by-step explanation:

a) using law of cosines: x is the side we do not know.

x² = 126² + 110² - 2.126.110. cos74°

x² = 20335.3

x = 142.6 ft

So 150 > 142.6, there is enough fance

b) using law of sine:

sin 74 / 142.6 = sinα/126 = sinβ/110

sin 74 / 142.6 = sinα/126

0.006741 = sinα/126

sinα = 0.849

α = sin⁻¹ (0.849)

α = 58.1°

sin 74 / 142.6 = sinβ/110

sin 74 / 142.6 = sinβ/110

0.006741 = sinβ/110

sinβ = 0.741

β = sin⁻¹ (0.741)

β = 47.9°

Checking: 74+58.1+47.9 = 180° ok

c) Using Heron A² = p (p-a) (p-b) (p-c)

p = a+b+c/2

p=126+110+142.6/2

p=189.3

A² = 189.3 (189.3-126) (189.3-110) (189.3-142.6)

A = 6661.5 ft²

1/3 A = 2220.5

So 2300 > 2220.5. The area you want to build is bigger than the area available.

The city will not approve