Please help me to solve the question below.
An electrical appliances shop owner estimates that his sales will increase by 5% every month.
The sales for the first month were $30,000. Find:
a)The sales for the 10th month
b)the total sales for the first year
To solve this question, we need to use the concept of compound interest, as the sales are increasing by 5% every month.
a) To find the sales for the 10th month, we need to calculate the sales value after 10 months.
We can use the formula A = P(1 + r)^n, where:
A is the final sales amount
P is the initial sales amount
r is the rate of growth per period
n is the number of periods
In this case, we have:
P = $30,000 (initial sales for the first month)
r = 5% = 0.05
n = 10 (number of months)
Using these values, we can calculate the sales for the 10th month:
A = $30,000(1 + 0.05)^10 = $30,000 * 1.05^10 ≈ $46,644.13
Therefore, the sales for the 10th month will be approximately $46,644.13.
b) To find the total sales for the first year, we need to calculate the cumulative sales amount for each month.
We can use the formula for the sum of a geometric series:
S = P[(1 - r^n)/(1 - r)], where:
S is the sum of the sales
P, r, and n have the same meaning as before
In this case, we have:
P = $30,000 (initial sales for the first month)
r = 5% = 0.05
n = 12 (number of months in a year)
Using these values, we can calculate the total sales for the first year:
S = $30,000[(1 - 0.05^12)/(1 - 0.05)] = $30,000 * (1 - 0.05^12) / (1 - 0.05) ≈ $372,748.79
Therefore, the total sales for the first year will be approximately $372,748.79.