Question

the average revenue of a tv manufacturing unit is given by $r(x) = 75x^2 - 5x^{\frac{1}{2}}$ where $x$ is the number of tvs sold by the firm. find the total revenue generated by the firm.

a. $75x^2 - 5x^{\frac{1}{2}}$

b. $75x^3 - 5x^{\frac{1}{2}}$

c. $75x^3 + 5x^2$

d. $75x^2 + 5x^{\frac{1}{2}}$

Explanation:

Step1: Recall Total Revenue Formula

Total revenue \( R(x) \) is average revenue \( r(x) \) multiplied by quantity \( x \), so \( R(x)=r(x)\times x \).

Step2: Substitute \( r(x) \) and Multiply

Given \( r(x) = 75x^{2}-5x^{\frac{1}{2}} \), then \( R(x)=(75x^{2}-5x^{\frac{1}{2}})\times x \).
Using the rule \( a^m\times a^n = a^{m + n} \), we get:
For the first term: \( 75x^{2}\times x=75x^{2 + 1}=75x^{3} \)
For the second term: \( - 5x^{\frac{1}{2}}\times x=-5x^{\frac{1}{2}+1}=-5x^{\frac{3}{2}} \)
So \( R(x)=75x^{3}-5x^{\frac{3}{2}} \)

Answer:

B. \( 75x^{3}-5x^{\frac{3}{2}} \)