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 . 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 
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.

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)

given that: $h(t) = -4.9t^2+24t+8$

a) at the top of the building, t=0, so 8 meters.

b) for maximum height, differentiate the given equation with respect to time and put it equal to zero. so,

$\frac{d(h(t))}{dt}=\frac{d(-4.9t^2+24t+8)}{dt}$

$=\frac{d(-4.9t^2)}{dt}+\frac{d(24t)}{dt}+\frac{d(8)}{dt}$

$\frac{d(h(t))}{dt}=-(2*4.9)t+24+0=-9.8t+24$

now put, $\frac{d(h(t))}{dt}=0$ then we get ---

-9.8t+24 = 0, $t=\frac{24}{9.8}=2.449$

thus, max time to reach at maximum height is at t = 2.449 sec

also the maximum height attain by the ball is $h(2.44...) = -4.9*(2.449)^2+24*2.449+8=37.388$

max height = 37.388 meters

3) Total brushels for 75 per acre = 75*20 = 1500

 

Suppose X is the number of plants added

 

The new number becomes 75 + X

 

Now each tree gives only 17 brushels

 

So the total number of brushels = (75 + X ) * 17 = 75*17 +17 X

 

This number has to be more than 1500 

 

1275 +17X should be grater than 1500

 

17 X should be greater than 225

 

X should be grater than = 13.28

 

But X can be integer only

So at least 14 trees must be added to each acre in addition to 75 to make it better deal in terms of fruit.