Unit2 - Work PDF

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1. Are the lines parallel, perpendicular, or neither? What is a rigorous algebraic solution for each problem? A. 3 3 34 4 4 3+ 4= 12 6 6 68 8 8− 6= 8+ 1 a. Upon taking the equations into the graphing calculator I am able to determine that the two equations are indeed parallel to each other. The example will be at the end (Fig. A) b. Now for the rigorous solution, you want to multiply the 3y+4x=12 by -2, and switch the equations up a bit. Our new equation will now be: -6y-8x=-24 -6y-8x=1 Because now, that they have the same slope, we can see that they are indeed parallel. 3 B. 3 + = 12 −8 8−= 8+ 1 a. Upon taking the equations into the graphing calculator I am able to determine that the two equations are indeed perpendicular to each other. The example will be at the end (Fig. B) b. Now, with this rigorus solution, we are going to prove that they do not have the same slope. So with this we will multiply the second equation by 3. So now we will have these equations: 3y+x=12 -3y-24x=3

With looking at that, we can conclude that the slopes are different. Slope of the first line is -1/3 where as the second line’s is -8. C. 4x-7y=10 7x+4y=1 a. Upon taking the equations into the graphing calculator I am able to determine that the two equations are indeed perpendicular to each other. The example will be at the end (Fig. C) b. So, while looking at this equation we can see it’s already set up how we would like it. We can also see what the slopes are. That being 4/7 and -7/4 making them complete opposites, therefore, perpendicular. 2. A ball is thrown from the top of a building. It’s height that’s given in meters above ground, as a function of time, in seconds, is given by: h(t)=-4.9t2+24t+8 What is the height of the building? What’s the maximum height the ball can reach? How long will it take to meet maximum height? A. h(0)=-4.9*02+24*0+8 =8 With the above equation, we can determine the height of the building is 8 meters. B. To figure out the maximum height, it’s best to figure out how long it takes to reach the maximum height. x=-b/2a=-24/2(-4.9) =-24/-9.8 =2.45

With the above equation, we can now see that it would take the ball 2.45 seconds to reach it’s maximum height. C. Now we can use the speed to figure out the maximum height. -4.9(2.45)2+24(2.45)+8 =37.40 With that equation, we now know the y-coordinate of the vertex of the parabola. The maximum height is 37.40 meters. (Fig. D) 3. Farmer figures out if she plants 75 trees per acre, each of the trees will yield 20 bushels of fruit. She estimates that each additional tree per acre, the yield would decrease by 3 bushels. How many trees should be planted per acre to maximize the harvest? B(n)=(20-3n)(75+n)A n = number of trees A= number of acres. A. So using the above equation, the graph shows that n=-34. (Fig. E, Blue) Now we take 75-34 =41 With that information, we can determine she should plant 41 trees, per acre. (Fig. E)

(Fig A.)

(Fig B.)

(Fig C.)

(Fig D)

(Fig E)

References: Graphing Calculator. (n.d.). Retrieved January 31, 2021, from https://www.desmos.com/calculator Abramson, J. P., Falduto, V., Gross, R., Lippman, D., Norwood, R., Rasmussen, M., . . . Fernandez, C. (2017). Chapter 3: Functions. In Algebra and trigonometry (pp. 279-356). Houston, TX: OpenStax....