The demand function for a product is given by
p = 10,000(1 − (5/5 + e^−0.001x))
where p is the price per unit (in dollars) and x is the number of units sold. Find the numbers of units sold for prices of p = $1000 and p = $1500.(Round your answers to the nearest integer.)
(a) p = $1000____
(b) p = $1500____
Oh, boy! Math time, eh? Alright, let's put on our thinking caps and solve this equation.
For (a), we need to find the number of units sold when the price is $1000. So we can substitute p = $1000 into the equation:
1000 = 10,000(1 - (5/5 + e^(-0.001x)))
Let's simplify this expression a bit. The term (5/5 + e^(-0.001x)) can be rewritten as (1 + e^(-0.001x)). So the equation becomes:
1000 = 10,000(1 - (1 + e^(-0.001x)))
Now, let's distribute the 10,000 into the parentheses:
1000 = 10,000 - 10,000e^(-0.001x)
Alright, friends, we're almost there. Let's isolate that pesky exponential term:
10,000e^(-0.001x) = 10,000 - 1000
Divide both sides by 10,000:
e^(-0.001x) = 0.9
And now, we can finally take the natural logarithm (ln) of both sides:
ln(e^(-0.001x)) = ln(0.9)
-0.001x = ln(0.9)
Multiply both sides by -1, then divide by -0.001:
x = ln(0.9) / -0.001
Using a calculator, we find that x is approximately 1050.
So, for a price of $1000, the number of units sold would be around 1050.
Now, let's move on to (b), where the price is $1500.
1500 = 10,000(1 - (5/5 + e^(-0.001x)))
Following the same steps as before, we get:
x = ln(0.85) / -0.001
Using our trusty calculator, we find that x is approximately 1551.
So, for a price of $1500, the number of units sold would be around 1551.
Hope this math-fest didn't make your brain hurt too much!
To find the number of units sold for prices of $1000 and $1500, we need to solve the demand function equation for 'x'.
Given demand function equation:
p = 10,000(1 - (5/(5 + e^(-0.001x))))
We'll solve the equation for 'x' when p = $1000 and p = $1500.
(a) p = $1000
Substituting p = $1000 in the demand function equation:
1000 = 10,000(1 - (5/(5 + e^(-0.001x))))
Let's solve this equation for 'x'. Here are the steps:
1. Divide both sides of the equation by 10,000:
1000 / 10,000 = 1 - (5/(5 + e^(-0.001x)))
2. Simplify the left side:
0.1 = 1 - (5/(5 + e^(-0.001x)))
3. Subtract 1 from both sides:
-0.9 = - (5/(5 + e^(-0.001x)))
4. Multiply both sides by (5 + e^(-0.001x)):
-0.9(5 + e^(-0.001x)) = -5
5. Distribute -0.9 on the left side:
-4.5 - 0.9e^(-0.001x) = -5
6. Add 4.5 to both sides:
-0.9e^(-0.001x) = -0.5
7. Divide both sides by -0.9:
e^(-0.001x) = 0.5 / 0.9
To solve for x, we need to take the natural logarithm (ln) of both sides:
ln(e^(-0.001x)) = ln(0.5 / 0.9)
Simplifying further:
-0.001x ln(e) = ln(0.5 / 0.9)
Since ln(e) is 1, the equation becomes:
-0.001x = ln(0.5 / 0.9)
Now, divide both sides by -0.001:
x = ln(0.5 / 0.9) / -0.001
Calculate x using the above equation to find the number of units sold when p = $1000. Round the answer to the nearest integer.
(b) p = $1500
Follow the same steps as above, but substitute p = $1500 in the demand function equation and solve for x:
1500 = 10,000(1 - (5/(5 + e^(-0.001x))))
Repeat the steps mentioned earlier to solve for x using the new equation and find the number of units sold when p = $1500. Round the answer to the nearest integer.