performing arithmetic of functions

Question

The profit of Johnny's Tires at time t is calculated by the function P(t) 169(3t +5)^1/7, where P is measured in millions of dollar. The Number of tires in production is calculated by N(t) 13(3t +5)^1/9. Determine a function for the profit per tire J(t) by finding (P/N)(t). Simplify your answer.

Answer 1

All we must do is create a fraction with P(t) in the numerator and N(t) in the denominator. So, we get (169*(3t+5)^1/7)/(13*(3t+5)^1/9). We need and can simplify this, though. So, let's separate the integers from the (3t+5)^n's.

So, we get 169/13*(3t+5)^1/7/(3t+5)^1/9. Let's divide 169/13 first, which is 13. So, so far, 13*(3t+5)^1/7/(3t+5)^1/9.

Now, we need to simplify (3t+5)^1/7/(3t+5)^1/9. Since the base numbers of the exponents in the numerators are the same, we can subtract their powers. To make it easier, we are going to change the powers, so that they have a common denominator. 1/7 becomes 9/63. 1/9 becomes 7/63.

So, we get (3t+5)^9/63/(3t+5)^7/63. Now let's subtract the exponents: (3t+5)^(9/63-7/63)=(3t+5)^2/63

In simplified form. we get 13*(3t+5)^2/63=J(t).

Answer 2

Hello,

(P/N)(t) is equivalent to [P(t)]/[N(t)]

In order to solve we divide the two functions and J(t) will be the quotient of that.

P(t)= 169(3t +5)^1/7

N(t)= 13(3t +5)^1/9

J(t)= [169(3t +5)^1/7] / [13(3t +5)^1/9]

By the properties of multiplication:

[169/13] * [(3t +5)^1/7] / [(3t +5)^1/9] is equivalent to [169(3t +5)^1/7] / [13(3t +5)^1/9]

We can then simplify and continue to solve

[169/13] * [(3t +5)^1/7] / [(3t +5)^1/9]

= [13] * [(3t +5)^1/7] / [(3t +5)^1/9]

By the properties of exponents:

(x^m) / (x^n) = x^(m-n)

Using the above property we can proceed to solve. However, since the fractions do not have like denominators we will have get equivalent fractions with like denominators.

Looking at 1/7 & 1/9 the lowest common multiple of the denominators 7 & 9 is 6. So I will multiple the first fraction 1/7 by 9/9 and I will multiple 1/9 by 7/7

that gives me 1/7 = 9/63 & 1/9 = 7/63

our expression now looks like

J(t) = [13] * [(3t +5)^9/63] / [(3t +5)^7/63]

= [13] * [(3t +5)^[(9/63) - (7/63)]

= [13] * [(3t +5)^(2/63)

J(t) = 13(3t +5)^(2/63)

I hope this helps.