University:
Shippensburg University of Pennsylvania
Course:
MAT 211 | Calculus I
Academic year:

2024
Views:

179

Pages:

3
Author:

Herbosambx

Question 1 Solve the equation and identify as a conditional equation, a contradiction, or an identity: 10 𝑚 + 18 − 7 𝑚 = 3 𝑚 + 25 m 10 +18− m 7 = m 3 +25 Answer Step 1: Given: 10 𝑚 + 18 − 7 𝑚 = 3 𝑚 + 25 m 10 +18− m 7 = m 3 +25 Step 2: Solving: 10 𝑚 − 7 𝑚 = 3 𝑚 + 25 − 18 m 10 − m 7 = m 3 +25−18 3 𝑚 = 3 𝑚 + 7 m 3 = m 3 +7 Simplify: 3 𝑚 + 144 = 3 𝑚 + 200 3m+144=3m+200 144 = 200 144=200 The given equation has no solution. Conclusion: 10 𝑚 + 18 − 7 𝑚 = 3 𝑚 + 25 m 10 +18− m 7 = m 3 +25 is a contradiction. Question 2 Find an equation of the conic described, graph the equation. Parabola: focus ( − 1 , 4.5 ) (−1,4.5), vertex ( − 1 , 3 ) (−1,3). Answer Step 1: The focus of the parabola is given as ( − 1 , 4.5 ) (−1,4.5). The vertex of the parabola is given as ( − 1 , 3 ) (−1,3). The axis of the parabola is given as 𝑥 = − 1 x=−1. Hence, the given parabola is a vertical parabola. Step 2: If the axis of the parabola is parallel to the y-axis and the vertex is ( ℎ , 𝑘 ) (h,k), the general equation of the parabola is given as: ( 𝑥 − ℎ ) 2 = 4 𝑎 ( 𝑦 − 𝑘 ) (x−h) 2 =4a(y−k) Here, 𝑎 = 4.5 − 3 = 1.5 a=4.5−3=1.5 Substitute: ( 𝑥 + 1 ) 2 = 4 ( 1.5 ) ( 𝑦 − 3 ) (x+1) 2 =4(1.5)(y−3) ( 𝑥 + 1 ) 2 = 6 ( 𝑦 − 3 ) (x+1) 2 =6(y−3) Simplify: 𝑥 2 + 2 𝑥 − 6 𝑦 + 19 = 0 x 2 +2x−6y+19=0 Final Answer: The required equation of the parabola is: 𝑥 2 + 2 𝑥 − 6 𝑦 + 19 = 0 x 2 +2x−6y+19=0 Question 3 Are these two lines parallel, perpendicular, the same line, or none of these? − 3 𝑥 = 3 𝑦 + 6 −3x=3y+6 4 𝑥 + 4 𝑦 = 2 4x+4y=2 Answer Two lines are parallel if their slopes are the same (or equal). Two lines are perpendicular if their slopes are negative reciprocals. Rewrite each line in slope-intercept form 𝑦 = 𝑚 𝑥 + 𝑏 y=mx+b: For − 3 𝑥 = 3 𝑦 + 6 −3x=3y+6: 𝑦 = − 𝑥 − 2 y=−x−2 For 4 𝑥 + 4 𝑦 = 2 4x+4y=2: 𝑦 = − 𝑥 + 1 2 y=−x+ 2 1 Since both lines have a slope of − 1 −1, they are parallel. Final Answer: (c) Parallel Question 4 Solve the equation or formula for the variable indicated. 10 𝑐 − 𝑓 = − 13 + 𝑐 𝑑 10c−f=−13+cd Solve for 𝑐 c. Answer 1: Subtract 𝑐 𝑑 cd from both sides: 10 𝑐 − 𝑓 − 𝑐 𝑑 = − 13 10c−f−cd=−13 Add 𝑓 f to both sides: 10 𝑐 − 𝑐 𝑑 = − 13 + 𝑓 10c−cd=−13+f Factor 𝑐 c on the left side: 𝑐 ( 10 − 𝑑 ) = − 13 + 𝑓 c(10−d)=−13+f Divide by ( 10 − 𝑑 ) (10−d): 𝑐 = − 13 + 𝑓 10 − 𝑑 c= 10−d −13+f Question 5 Solve each system of equations using substitution. State whether the system is inconsistent or consistent with a dependent equation. 𝑦 = 4 𝑥 − 1 y=4x−1 8 𝑥 − 2 𝑦 = 2 8x−2y=2 Answer Step 1: An inconsistent system has no solutions. A consistent system has at least one solution. Step 2: We have: 𝑦 = 4 𝑥 − 1 (1) y=4x−1(1) Substitute into: 8 𝑥 − 2 𝑦 = 2 8x−2y=2 8 𝑥 − 2 ( 4 𝑥 − 1 ) = 2 8x−2(4x−1)=2 8 𝑥 − 8 𝑥 + 2 = 2 8x−8x+2=2 Conclusion: The two equations represent the same line. Hence, the system is consistent and dependent.