These questions are pertaining to Applied Calculus

Question

The demand equation for a certain product is given by p= 110 - 0.025x, where p is the unit price ( in dollars) of the product and x is the number of units produced. The total revenue obtained by producing and selling x units is given by R = xp.

 

Determine prices p that would yield a revenue of 9,980 dollars.

 

Lowest such price =

Highest such price=

 

2) Find b and c so that y= -3x^2 + bx + c has a vertex (-10,7)

 

b=

c=

Answer 1

Hi Keissa,

 

1.

 

Given that R = xp and p = 100-0.025x, you must substitute p into the R equation.

 

R = xp = x(100-0.025x).

 

We are interested in when R = 9980.

 

9980 = x(100 - 0.025x) = 100x-0.025x^2

 

or equivalently,

 

0.025x^2-100x+9980 = 0

 

Now, you must solve this quadratic equation either by using the quadratic formula or by letting y=0.025x^2-100x+9980 on a graphing calculator and finding the zeros (x-intercepts). This will give you the two x values, but keep in mind that the question asks for the prices.

 

2.

 

You need to know that the vertex form of a quadratic is given by y = a(x-h)^2+k where

a = scaling factor

The point (h,k) is that vertex.

 

So since (-10,7) is the vertex,  h = -10, k = 7.

 

Therefore  y = a(x+10)^2+7.

 

But this must also equal y = x^2+bx+c.

 

Setting these two equal to each other, we get a(x+10)^2+7  = x^2+bx+c

 

Now let's expand the left hand side: a(x+10)(x+10)+7 = a(x^2+20x+100)+7  = ax^2+20ax+100a+7.

 

This is what we end up with: ax^2+20ax+100a+7 = x^2+bx+c

 

Matching coefficients, we get: ax^2=x^2, 20ax = bx, 100a+7 = c.

 

Go ahead and finish from here.

 

I hope this helps! If you need additional help in this course, please reach out to me to setup online sessions. I'd be more than willing to help you exceed your goals!

 

Kind regards,

 

Anthony