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UNIT 3 : QUADRATIC FUNCTIONS & EQUATIONS
LESSON
3: PROBLEM SOLVING WITH QUADRATIC
FUNCTIONS HOMEWORK QUESTIONS
2. A ball is thrown
in the air from ground level. Its height
in metres at any time t seconds is given by h(t) = -4.9t2 + 24.5t + 0.4.
a) Find the initial height of the ball.
b) Find the maximum height attained by the ball and when it occurs.
c) When will the ball hit the ground.
d) Draw the graph showing time on the x-axis.
3. Laura and Paul have 48 yds of fencing to enclose a rectangular garden in their yard. One side of the garden will be against the house.
Find the dimensions they should use to maximize the area of the garden.
4. Farmer Al has a 90 lb hog which gains an average of 2.5 lb/day. However, the present price for pork is $1.40/lb and is falling at $0.02/day.
It costs him $ 0.60/day to maintain the hog. How many days should he wait before he sells the hog in order to maximize profits?
5. A church window consists of a semi-circular section of radius x on top of a clear rectangular section of width 2x. Find the dimensions of the window
that admits the most light if the perimeter of the entire window is 15 ft.
6. Find the number which exceeds its square by the greatest possible amount.
7. The demand function for Power Pencil’s mechanical pencil set is p(x) = -4x + 15 where x is the number of sets sold in thousands and p(x) is
the price of one set in dollars. The Cost function is given by C(x) = 3x + 6
a) Find the Revenue and Profit functions .
b) Determine the price that will maximize profit.
c) Sketch the graph of the profit function.
d) What is the break-even point for this product
2. A ball is thrown
in the air from ground level. Its
height in metres at any time t seconds is given by h(t) = -4.9t2 + 24.5t +
0.4.
a) Find the initial height of the ball.
b) Find the maximum height attained by the ball and when it occurs.
c) When will the ball hit the ground.
d) Draw the graph showing time on the x-axis.
Solution: (answers given to one
decimal place)
a) Initial height occurs when t = 0
h(0) = -4.9(0)2 +24.5(0) + 0.4
= 0.4
Therefore the initial height is 0.4 m. Note – this will be the y – intercept (h – intercept in this case) on the graph.
b) Complete the square to find the maximum height attained
c) When the ball hits the ground the height will be zero. Hence let h = 0 in the function and solve the resulting quadratic equation.
d)
3. Laura and Paul have 48 yds of fencing to enclose a rectangular garden in their yard. One side of the garden will be against the house.
Find the dimensions they should use to maximize the area of the garden.
Solution:
à Determine
what is unknown or what you are asked to find?
Assign variables to the unknowns.
Here the
dimensions of the field are unknown.
Let the length be x and the width be y.
Side of house
y y y
x
à
Determine the quantity to be maximized or minimized.
We are asked to maximize the area. Hence our formula is:
à Write this formula as a function of
one variable.
Since the available fencing is 48 yds, we have a secondary relation between the two variables
4. Farmer Al has a 90 lb hog which gains an average of 2.5 lb/day. However, the present price for pork is $1.40/lb and is falling at $0.02/day.
It costs him $ 0.60/day to maintain the hog. How many days should he wait before he sells the hog in order to maximize profits?
Solution:
Profit is found by subtracting costs (expenses)
from revenue or income.
Let the number of days he waits be
x.
After
“x” days the new price will be (1.40 – 0.02x) as he loses $0.02/day
The weight
of the hog will rise to (90 + 2.5x) as
it gains 2.5 lb/day.
Hence
the revenue he receives from the sale of the hog is Quantity x
Price = (90 +
2.5x)(1.40 – 0.02x)
The cost
of maintaining the hog is 0.60x [$0.60/day]
Profit
= Revenue – Cost
5. A church window consists of a semi-circular section of radius x on top of a clear rectangular section of width 2x. Find the dimensions of the window
that admits the most light if the perimeter of the entire window is 15 ft.
6. Find the number which exceeds its square by the greatest possible amount.
Solution:
à
Determine what is unknown or what you are asked to find? Assign variables to the unknowns.
Let the number be x.
à
Determine the quantity to be maximized or minimized.
à
Complete the square to maximize this function
7. The demand function for Power Pencil’s mechanical pencil set is p(x) = -4x + 15 where x is the number of sets sold in thousands and p(x) is
the price of one set in dollars. The Cost function is given by C(x) = 3x + 6
a) Find the Revenue and Profit functions .
b) Determine the price that will maximize profit.
c) Sketch the graph of the profit function.
d) What is the break-even point for this product
Solution:
a) Recall our
Revenue and Profit formulas (see lesson for formulas) 
c)
b) The price will be p(x) = p(1.5) = -4(1.5) + 15 = $9.00/set to maximize profit
d) Break even occurs when the profit = 0 or where revenues exactly equal costs. These are the x-intercepts of the graph.
