Please I kindly need help in solving these qustions

Question

1. Determine whether the lines given by the equations below are parallel, perpendicular, or neither. Also, find a rigorous algebraic solution for each problem.

a.    b.    c. 

 [Suggestion: go to www.desmos.com/calculator, write the two equations and try to conclude the answer.]

2. A ball is thrown in the air from the top of a building. Its height, 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 is the maximum height reached by the ball? How long does it take to reach maximum height? Also, find a rigorous algebraic solution for the problem.

[Suggestion: go to www.desmos.com/calculator and write 

y= -4.9x² + 24x + 8

and observe the answers.]

3. A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest? Also, find a rigorous algebraic solution for the problem.

[Suggestion: finding a function B(n), that is, the number of bushels corresponding to n trees per acre, go to the www.desmos.com/calculator and try to use a graph to visualize the situation and test your formula.]

Answer 1

1)

a)

3y + 4x = 12, has slope -4/3

-6y = 8x+1 has slope -8/6 = - 4/3

the slopes are the same so the lines are parallel

b) 3y +x = 12, has slope -1/3

y = 8x+1 has slope 8

the slopes are different, not negative reciprocals,

so the lines are neither parallel nor perpendicular;

they will intersect at a unique point

c) 4x - 7y = 10 , has slope 4/7

7x + 4y = 1 has slope -7/4

the slopes are negative reciprocals, so the lines are

perpendicular

2) at the top of the building, t=0, so 8 feet

max occurs at -b/2a = -24/(2*-4.9) = 2.448979591835...

h(2.44...) = 37.38775

3) f(75) = 25

f(78) = 28

f(81) = 31

slope is 1.

y = x+ 50 is the linear equation.