To calculate the volume of a chemical produced in a day a chemical manufacturing company uses the following formula below: This is for SAT prep so an explanation for the questions would be appreciated

Question

To calculate the volume of a chemical produced in a day a chemical manufacturing company uses the following formula below:

V(x)=[C₁(x)+C₂(x)](H(x))

where x represents the number of units produced. This means two chemicals are added together to make a new chemical and the resulting chemical is multiplied by the expression for the holding container with respect to the number of units produced. The equations for the two chemicals added together with respect to the number of units produced are given below:

C₁(x)=x/(x+1), C₂(x)=2/(x-3)

The equation for the holding container with respect to the number of units produced is given below:

H(x)=(x³-9x)/x

a. What rational expression C(x) do you get when you combine the two chemicals?

b. What is the simplified equation of V(x)?

c. What would the volume be if 50, 100, or 1000 units are produced in a day?

d. The company needs a volume of 3000 c³ How many units would need to be produced in a day?

Answer 1

If I'm reading this correctly...

part (a) is asking you to simplify C₁(x) + C₂(x) into a new function C(x)

Here we have C₁(x)=x/(x+1), C₂(x)=2/(x-3)

so C(x) = x/(x+1) + 2/(x-3)

to add fractions with different denominators we can rely on the following formula

a/b + c/d = (ad + bc)/bd

so C(x) is further simplified to (x(x-3) + 2(x+1))/((x+1)(x-3)) = (x²-x+2)/((x+1)(x-3)) = ((x-2)(x+1))/((x+1)(x-3))

and finally C(x) = (x-2)/(x-3)

part (b) is asking next to simplify V(x) = C(x)H(x). Let's first simplify H(x)

H(x) = (x³-9x)/x = x²-9 = (x+3)(x-3)

plugging in C and H, we get:

V(x) = ((x-2)/(x-3))(x+3)(x-3) = (x-2)(x+3) = x²+x-6

part (c) asks you to plug in some values of x to solve for the volume V(x)

V(50) = 48*53 = 2,544

V(100) = 98*103 = 10,094

V(1000) = 998*1003 = 1,000,994

part (d) asks you to solve for x given a particular volume V(x)

V(x) = 3000 = x²+x-6

x²+x-3006 = 0

x = (-1 ± √12025)/2 = 54.33 (chemical units)