University:
Morehouse College
Course:
HMTH 105 | College Algebra 1
Academic year:

2024
Views:

432

Pages:

4
Author:

Reynaldo Wong

Question 1 A plane flies directly over a radar station while traveling at 500 mi/h 500mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. Answer 1 Given the diagram: 𝑥 2 + 1 2 = 𝑦 2 x 2 +1 2 =y 2 Differentiate with respect to 𝑡 t: 2 𝑥 𝑑 𝑥 𝑑 𝑡 + 0 = 2 𝑦 𝑑 𝑦 𝑑 𝑡 2x dt dx +0=2y dt dy 𝑑 𝑥 𝑑 𝑡 = 𝑦 𝑥 𝑑 𝑦 𝑑 𝑡 dt dx = x y dt dy We are given that 𝑑 𝑥 𝑑 𝑡 = 500 mi/h dt dx =500mi/h. We also need to find 𝑥 x at the moment 𝑦 = 2 mi y=2mi: 𝑥 2 + 1 2 = 2 2 𝑥 2 = 4 − 1 = 3 𝑥 = 3 ( ignore negative root ) x 2 +1 2 =2 2 x 2 =4−1=3x= 3 (ignore negative root) Substitute into the equation: 𝑑 𝑦 𝑑 𝑡 = 𝑥 𝑦 𝑑 𝑥 𝑑 𝑡 dt dy = y x dt dx 𝑑 𝑦 𝑑 𝑡 = 3 2 ⋅ 500 dt dy = 2 3 ⋅500 𝑑 𝑦 𝑑 𝑡 = 250 3 mi/h dt dy =250 3 mi/h Question 2 A consumer in a grocery store pushes a cart with a force of 35 N 35N directed at an angle of 2 5 ∘ 25 ∘ below the horizontal. Find the work done by the shopper as she moves down a 50.0 m 50.0m length aisle. Answer 1 Given: 𝐹 app = 35 N , 𝜃 = 2 5 ∘ , Δ 𝑥 = 50 m F app =35N,θ=25 ∘ ,Δx=50m 𝑊 = 𝐹 app cos ⁡ 𝜃 ⋅ Δ 𝑥 W=F app cosθ⋅Δx 𝑊 = 35 ⋅ cos ⁡ ( 2 5 ∘ ) ⋅ 50 W=35⋅cos(25 ∘ )⋅50 𝑊 ≈ 1586.04 J W≈1586.04J Question 3 Describe how to identify the vertex angle of an isosceles triangle. Answer 1 The vertex angle of an isosceles triangle is the angle opposite the base of the triangle. (Recall that the two congruent angles are opposite the legs.) Question 4 What is cot ⁡ 0 cot0? Answer 1 It isn’t defined, as tan ⁡ 0 = 0 tan0=0 and cot ⁡ 𝜃 = 1 tan ⁡ 𝜃 cotθ= tanθ 1 . cot ⁡ 0 = 1 0 = ∞ cot0= 0 1 =∞ Question 5 For an acute angle 𝐴 A, can cos ⁡ 𝐴 cosA be larger than 1? Explain. Answer 1 The cosine of an angle is the ratio of the side length adjacent to the angle to the hypotenuse in a right triangle ( cos ⁡ = adj hyp cos= hyp adj ). Since the hypotenuse is always the longest side, hyp > adj hyp>adj, which means the cosine ratio is always less than 1. Hence, cos ⁡ 𝐴 cosA cannot be larger than 1.