a small car - detailing business has the following details: the cost to detail each car is $15.00. the fixed…

a small car - detailing business has the following details: the cost to detail each car is $15.00. the fixed costs are $50.00 per month. the total expense function is given by e(q)=50 + 15q where q is the number of cars detailed and e(q) is the total expense. the business owner wants to create a spreadsheet to calculate and track monthly expenses based on the number of cars detailed. the spreadsheet has this configuration: a: number of cars detailed (q), b: total expense (e). to fill out the total expense column, the business owner needs to input the formula for calculating total expenses in b2:b10. if the business owner wants to sum the total expenses for detailing all cars from 0 to 8, which value should be the result of the sum? 850, 990, 720, 765

a small car - detailing business has the following details: the cost to detail each car is $15.00. the fixed costs are $50.00 per month. the total expense function is given by e(q)=50 + 15q where q is the number of cars detailed and e(q) is the total expense. the business owner wants to create a spreadsheet to calculate and track monthly expenses based on the number of cars detailed. the spreadsheet has this configuration: a: number of cars detailed (q), b: total expense (e). to fill out the total expense column, the business owner needs to input the formula for calculating total expenses in b2:b10. if the business owner wants to sum the total expenses for detailing all cars from 0 to 8, which value should be the result of the sum? 850, 990, 720, 765

Answer

Explanation:

Step1: Recall the expense - function

The total expense function is $E(q)=50 + 15q$, where $q$ is the number of cars detailed.

Step2: Calculate the sum of expenses for $q$ from 0 to 8

We need to find $\sum_{q = 0}^{8}(50+15q)$. First, split the sum into two parts using the property $\sum_{i = 1}^{n}(a + b_i)=na+\sum_{i = 1}^{n}b_i$. So $\sum_{q = 0}^{8}(50+15q)=\sum_{q = 0}^{8}50+15\sum_{q = 0}^{8}q$. The sum $\sum_{q = 0}^{8}50$ is just $50\times9$ (since there are 9 values of $q$ from 0 to 8), and $\sum_{q = 0}^{8}q=\frac{8\times(8 + 1)}{2}=36$ (using the formula $\sum_{i = 1}^{n}i=\frac{n(n + 1)}{2}$, and here we consider from 0 to 8 which has 9 terms for the constant part and the sum of integers from 1 to 8 for the $q$ - part).

Step3: Calculate the two - part sum

$\sum_{q = 0}^{8}50=50\times9 = 450$ and $15\sum_{q = 0}^{8}q=15\times36=540$. Then $\sum_{q = 0}^{8}(50 + 15q)=450+540=990$.

Answer:

990