Question #183589

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

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. 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.

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.

Expert's answer

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)

h(t) = -4.9t^{2} + 24t + 8,

h=h(t=0)=8 (m),

h'=-9.8t+24,

h'=0,

t=24/9.8=2.45 (s),

l=h(t=2.45)=37.4 (m),

3)

f(x)=(75+x)(20-3x),

f'(x)=20-3x-3(75+x)=-205-6x,

f'(x)=0,

x≈-34,

number of trees is 75-34=41,

number of bushels of fruit is (75-34)*(20-3*(-34))=41*122=5002.

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