The question is-
Aki’s Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x) = 0.6x^2-0.8x+10.618, where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
I think- the function, C(x), gives the average cost when x hundred bicycles are made. This is my quadratic function right? Where the coefficient of x^2 is positive? Then the minimum value will be at the vertex?
So I need the form y=a(x-h)^2+k to get my minimum value- x=h.
Now, I need to separate the constant part from the remainder of the function and factor the coefficient of x^2 from the terms containing x. (I am supposed to round to three decimal places if needed.
C(x) = 0.6x^2-0.8+10.618
Now I am lost
C(x)= 0.6 (x^2-?.??x)+10.618
Where does this equation come from?
With that I can complete the square by taking half the answer and squaring it.
Yes, C(x) is the quadratic cost function, and the minimum is at the vertex.
C(x) = 0.6x^2-0.8x+10.618
= 0.6(x^2 - 1.333x + 17.697)
= 0.6 (x^2 - 1.333x +0.444 + 17.253)
= 0.6 [(x- 0.667)^2 + 17.53]
There is a minimum average cost when x = 0.667 hundred bikes (or 67 bikes).
These must be pretty expensive bikes. The minimum cost is 10.35 hundred dollars ($1035)
How do you get 1.333x+17.697 and 1.333x+0.444+17.253 and (x-0.667)^2+17.53?
By dividing out 0.6 and then completing the square. Check the numbers. They work.
one of two positive intergers is 5 less than the other. if the product of the two intergers is 24, find the intergers
x*y = 24
y = x - 5 so the equation becomes:
x*(x - 5) = 24
expand: x^2 - 5x - 24 - 0
Factor by sum and product method:
(x - 8)(x + 3) = 0 so x = 8 or x = -3
since the integers are positive, the solution is x = 8, and y = x - 5, so
y = 8 - 5, y = 3
the two integers are 3 and 8
can you help me
Find 2 numbers with a difference of 10 and a product that is a minimum. Find the minimum product.
243548696797-
5654353574768
756746
To find the minimum value of the average cost per bicycle, you're on the right track by using the vertex form of the quadratic function. Let's continue the steps to complete the square:
C(x) = 0.6x^2 - 0.8x + 10.618
First, let's factor out the coefficient of x^2, which is 0.6:
C(x) = 0.6(x^2 - (0.8/0.6)x) + 10.618
Now we need to complete the square inside the parentheses. To do this, take half the coefficient of x (which is -0.8/0.6) and square it:
(-0.8/0.6)^2 = 0.8889
Now add and subtract this value inside the parentheses:
C(x) = 0.6(x^2 - (0.8/0.6)x + 0.8889 - 0.8889) + 10.618
Now, let's rewrite the expression inside the parentheses as a perfect square trinomial:
C(x) = 0.6((x - 0.8/0.6)^2 - 0.8889) + 10.618
Simplifying further:
C(x) = 0.6(x - 1.3333)^2 - 0.5334 + 10.618
Finally, combine the constant terms:
C(x) = 0.6(x - 1.3333)^2 + 10.0846
Now the quadratic function is in the form y = a(x - h)^2 + k, with the vertex at (h, k). In this case, the minimum value occurs at the vertex, which is (1.3333, 10.0846). Rounding to three decimal places, the shop should build approximately 1.333 hundred bicycles to minimize the average cost per bicycle.