Could someone help, please?

The sales \( S \) (in thousands of units) of a cleaning solution after \( X \) hundred dollars is spent on advertising are given by:
\[ S = 10(1 - e^{kx}) \]

Now, when $325 is spent on advertising, 2300 units are sold.

(a) Complete by solving for \( k \). (Round your answer to 4 decimal places)
\[ k = \_\_\_\_\_\_ \]

(b) What number of units will be sold if advertising is raised to $425?
\[ \_\_\_\_\_\_ \text{ Units} \]

2.3 = 10 ( 1 − e 3.25 k )) 2.3 = 10 − 10 e 3.25 k 10 e 3.25 k = 7.7 e 3.25 k = 0.77 3.25 k = ln 0.77 k = 3.25 ln 0.77 ​ k ≈ − 0.0804
S = 10 ( 1 − e 3.25 l n 0.77 ​ ⋅ 4.25 )
Calculating this is a pain in the ***, but as a result you get approximately 2.8949 which in thousands of units is 2894 (rounded down since you can't have a fraction of a unit).

Answered by konrad509 | 2024-06-10

(a) The value of k is -0.0257 .
(b) The number of units that will be sold if advertising is raised up to $425 1.034 units.
To solve this problem, we use the given equation for sales S: S = 10 ( 1 − e k x ) .
(a) First, we solve for the constant k. Given that when $325 is spent on advertising, 2300 units are sold, we substitute these values into the equation:
S = 2300 / 1000 = 2.3 and x = 325 / 100 = 3.25
Therefore, we have:
2.3 = 10 ( 1 − e 3.25 k )
Next, solve for e 3.25 k :
2.3 / 10 = 1 - e 3.25 k
0.23 = 1 - e 3.25 k
-0.77 = - e 3.25 k
e 3.25 k = 0.77
Taking the natural logarithm on both sides:
3.25k = ln(0.77)
k = ln(0.77) / 3.25
Using a calculator, we find:
k ≈ -0.0835203498 / 3.25
k ≈ -0.0257
So, k = -0.0257 (rounded to four decimal places).
(b) To find the number of units sold when $425 is spent on advertising, we substitute x = 4.25 into the sales equation:
S = 10 ( 1 − e 4.25 \t ×− 0.0257 )
Calculating the exponent:
-0.0257 × 4.25 ≈ -0.109225
e − 0.109225 ≈ 0.8966
Now, substitute back:
S = 10(1 - 0.8966)
S = 10 × 0.1034
S ≈ 1.034
Therefore, the number of units sold when $425 is spent on advertising is 1,034 units.

Answered by ToddBoyce | 2024-07-04

We calculated the value of k as approximately -0.0804 when $325 is spent on advertising. We also found that with an increase to $425 in advertising, approximately 2888 units would be sold. These calculations were derived using the given sales formula for the cleaning solution.
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Answered by konrad509 | 2024-09-27