An advertising agency conducted a survey and found that the number of units sold, N, is related to the amount a spent on advertising (in dollars) by the following formula:
N = 1,600 + 300 ln a (a ≥ 1).
How much needs to be spent (to the nearest hundred dollars) to sell 4,000 units?
4000 = 1600 + 300 ln a
ln a = 8
a = e^8 = 2980 or 3000
To find out how much needs to be spent to sell 4,000 units, we can use the given formula: N = 1,600 + 300 ln a, where N is the number of units sold and a is the amount spent on advertising.
In this case, we want to find the value of a when N is 4,000. So, we can substitute N = 4,000 into the formula and solve for a:
4,000 = 1,600 + 300 ln a
To isolate ln a, we subtract 1,600 from both sides:
4,000 - 1,600 = 300 ln a
2,400 = 300 ln a
Next, we divide both sides by 300:
2,400 / 300 = ln a
8 = ln a
Now, to find the value of a from the natural logarithm (ln), we can raise e, the base of the natural logarithm, to the power of both sides of the equation:
e^8 = e^(ln a)
By the properties of logarithms, the exponential function e^(ln x) simply equals x:
a = e^8
Using a calculator or a math software, we can approximate e^8 to the nearest hundred dollars:
a ≈ 2,980.957
Therefore, to sell 4,000 units, approximately $2,981 (to the nearest hundred dollars) needs to be spent on advertising.