+895 1) Find the polynomial function in factored form: \(y=a(x-b)(x-c)(x-d)\) for the given data.
| x | 1 | 3 | -4 | -1 |
| y | 0 | 0 | 0 | 2 |
2) The Sunspot Small Appliance Company determines that the supply function for their EverCurl hair dryer is \(S(p)=6+0.001p^3\) and the demand function is \(D(p)=80-0.02p^2\), where p is the price of the hair dryer. Determine the price for which the supply is equal to the demand and the nuumber of hair dryers corresponding to this equillibrium price.
+9479 1) When y = 0 , x = 1, 3, or -4
A polynomial in that form with zeros at 1, 3, and -4 is
y = a(x - 1)(x - 3)(x + 4)
To find a , plug in -1 for x and 2 for y .
2 = a(-1 - 1)(-1 - 3)(-1 + 4)
2 = a(-2)(-4)(3)
2 = 24a
a = 1/12
So the polynomial is y = (1/12)(x - 1)(x - 3)(x + 4) And here's a graph to check it.
2) We want to know what p is when supply = demand . That is, when
6 + 0.001p3 = 80 - 0.02p2
Let's solve this using a graph. The p value that makes the equations have the same y value is
p ≈ 36.27
Look at the graph or plug this value for p into either function to find out how many hair dryers are supplied and demanded at this price.
D( 36.27 ) = 80 - 0.02( 36.27 )2 ≈ 53
+9479 1) When y = 0 , x = 1, 3, or -4
A polynomial in that form with zeros at 1, 3, and -4 is
y = a(x - 1)(x - 3)(x + 4)
To find a , plug in -1 for x and 2 for y .
2 = a(-1 - 1)(-1 - 3)(-1 + 4)
2 = a(-2)(-4)(3)
2 = 24a
a = 1/12
So the polynomial is y = (1/12)(x - 1)(x - 3)(x + 4) And here's a graph to check it.
2) We want to know what p is when supply = demand . That is, when
6 + 0.001p3 = 80 - 0.02p2
Let's solve this using a graph. The p value that makes the equations have the same y value is
p ≈ 36.27
Look at the graph or plug this value for p into either function to find out how many hair dryers are supplied and demanded at this price.
D( 36.27 ) = 80 - 0.02( 36.27 )2 ≈ 53
+129852 .001p^3 + 6 = -.02p^2 + 80
p^3 + 6000 = -20p^2 + 80000
p^3 + 20p^2 - 74000 = 0
Note, AT.......there IS a "formula" for finding cubic root(s)...but...it's EXTREMELY messy....!!!!!
The "actual" real-number answer is :
p = 10/3 (-2 + (991 - 3 sqrt(109113))^(1/3) + (991 + 3 sqrt(109113))^(1/3))
Let's just go with hectictar's solution...!!!!!
+895 After seeing that long and obnoxious answer, I think my teacher will be okay with me using my graphing calculator.
Thanks CPhill!!!
