Question

generally, the more money a company spends on advertising, the higher the sales. let a represent the amount of money spent on advertising (in $100s). then the amount of money in sales s(a) (in $1000s) is given by s(a) = 10 + 19 log(a + 1) where a ≥ 0. part: 0 / 4 part 1 of 4 (a) the value of s(1) ≈ 15.7 means that if $100 is spent on advertising, $15,700 is returned in sales. find the values of s(15), s(23), and s(32). round to one decimal place. interpret the meaning of each function value in the context of this problem. s(15) ≈ this means that if the company spends $1500 on advertising, $ is returned in sales.

Explanation:

Step1: Substitute \( a = 15 \) into \( S(a) \)

We have the function \( S(a)=10 + 19\log(a + 1) \). Substitute \( a = 15 \) into it:
\( S(15)=10+19\log(15 + 1)=10+19\log(16) \)
First, calculate \( \log(16) \approx 1.2041 \) (assuming base 10 logarithm, if it's natural logarithm, \( \ln(16)\approx2.7726 \), but from the example \( S(1)\approx15.7 \), let's check with base 10: \( S(1)=10 + 19\log(2)\approx10+19\times0.3010 = 10 + 5.719=15.719\approx15.7 \), so it's base 10 logarithm).
So \( 19\log(16)\approx19\times1.2041 = 22.8779 \)
Then \( S(15)=10 + 22.8779=32.8779\approx32.9 \) (rounded to one decimal place)

Step2: Interpret the value

Since \( a \) is in $100s, when LXI1, the advertising cost is LXI2. And \( S(a) \) is in $1000s, so the sales amount is \( S(15)\times1000 \). Since \( S(15)\approx32.9 \), the sales amount is \( 32.9\times1000=\$32900 \)

Answer:

\( S(15)\approx\boldsymbol{32.9} \)
This means that if the company spends $1500 on advertising, $ \boldsymbol{32900} is returned in sales.