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

2024
Views:

47

Pages:

2
Author:

Reynaldo Wong

Question 1 Find the point on the line 𝑦 = 5 𝑥 + 3 y=5x+3 that is closest to the origin. Answer 1 Let ( 𝑥 , 𝑦 ) (x,y) be the point on the line that is closest to the origin. Since 𝑦 = 5 𝑥 + 3 y=5x+3, so the point ( 𝑥 , 𝑦 ) (x,y) on the line becomes ( 𝑥 , 5 𝑥 + 3 ) (x,5x+3), so now we use the distance formula and minimize it. Distance between ( 0 , 0 ) (0,0) and ( 𝑥 , 5 𝑥 + 3 ) (x,5x+3) is given by: 𝐷 = ( 𝑥 − 0 ) 2 + ( 5 𝑥 + 3 − 0 ) 2 D= (x−0) 2 +(5x+3−0) 2 𝐷 = 𝑥 2 + ( 25 𝑥 2 + 30 𝑥 + 9 ) D= x 2 +(25x 2 +30x+9) 𝐷 = 26 𝑥 2 + 30 𝑥 + 9 D= 26x 2 +30x+9 Now to minimize the distance, we find the critical point by setting 𝐷 ′ = 0 D ′ =0. Here using the chain rule, derivative is: 𝐷 ′ = 1 2 ( 26 𝑥 2 + 30 𝑥 + 9 ) − 1 2 ⋅ 𝑑 𝑑 𝑥 ( 26 𝑥 2 + 30 𝑥 + 9 ) D ′ = 2 1 (26x 2 +30x+9) − 2 1 ⋅ dx d (26x 2 +30x+9) = 1 2 26 𝑥 2 + 30 𝑥 + 9 ( 52 𝑥 + 30 ) = 2 26x 2 +30x+9 1 (52x+30) ⇒ 𝐷 ′ = 52 𝑥 + 30 2 26 𝑥 2 + 30 𝑥 + 9 ⇒D ′ = 2 26x 2 +30x+9 52x+30 Now 𝐷 ′ = 0 D ′ =0: 52 𝑥 + 30 = 0 52x+30=0 𝑥 = − 30 52 = − 15 26 x=− 52 30 =− 26 15 Question 2 Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is 6 0 ∘ 60 ∘ , how far are you from the base of the plateau? Answer 1 Let 𝑥 x = the distance from the base of the plateau to you. Given the length of the side opposite the angle, we can use the tangent ratio to write: tan ⁡ 6 0 ∘ = opposite adjacent = 50 𝑥 tan60 ∘ = adjacent opposite = x 50 𝑥 = 50 tan ⁡ 6 0 ∘ x= tan60 ∘ 50 𝑥 ≈ 28.9 m x≈28.9m Question 3 The leg of a right triangle that lies on one ray of angle 𝜃 θ is called the ? leg, and the leg that lies across triangle from 𝜃 θ is called the ? leg. Answer 1 The leg of a right triangle that lies on one ray of angle 𝜃 θ is called the adjacent leg, and the leg that lies across triangle from 𝜃 θ is called the opposite leg. Question 4 Draw a triangle that satisfies the set of conditions. Then classify the triangle. A triangle with three acute angles and three congruent sides. Answer 1 Having three congruent sides, we use the same tick marks to represent this. The angle formed are acute using any side length. The triangle is equilateral. Question 5 Tell whether two angles can be as described. Justify your answers. a. vertical and complementary b. vertical and supplementary c. complementary and supplementary Answer 1 a. Yes, if the intersecting lines form two 4 5 ∘ 45 ∘ angles as vertical angles, then they are also complementary as they add up to 9 0 ∘ 90 ∘ . b. Yes, if the intersecting lines are perpendicular then the vertical angles are also supplementary as they add up to 18 0 ∘ 180 ∘ . c. No, two complementary angles have a sum of 9 0 ∘ 90 ∘ while two supplementary angles have a sum of 18 0 ∘ 180 ∘ . This cannot happen at the same time.