I can't calculate 1 as the answer given to the following problem. I keep getting 2. Can you show me where am I going wrong?
The spread of rumor among 100 people can be approximated by the equation log10(x) – log10(100-x) = (0.1)t – log10(90), where x is the number of people who have heard the rumor after t days. How many people have heard the rumor after 10 days?
To solve the problem, we first need to rearrange the given equation:
log10(x) - log10(100-x) = (0.1)t - log10(90)
Simplifying the equation further, we can use the logarithmic property log(a) - log(b) = log(a/b):
log10(x/(100-x)) = (0.1)t - log10(90)
Next, we can raise both sides of the equation to the power of 10 to remove the logarithms:
10^(log10(x/(100-x))) = 10^((0.1)t - log10(90))
Simplifying, we have:
x/(100-x) = 10^((0.1)t) / 90
Now, we can cross-multiply to further simplify:
x * 90 = (100-x) * 10^((0.1)t)
Expanding and rearranging:
90x = 1000^((0.1)t) - 10^((0.1)t)x
Combining like terms:
10^((0.1)t)x + 90x = 1000^((0.1)t)
Factoring out the common factor, x:
x(10^((0.1)t) + 90) = 1000^((0.1)t)
Finally, dividing both sides by (10^((0.1)t) + 90), we get:
x = 1000^((0.1)t) / (10^((0.1)t) + 90)
To find the number of people who have heard the rumor after 10 days, substitute t=10 into the equation:
x = 1000^((0.1)*10) / (10^((0.1)*10) + 90)
Evaluate the right-hand side of the equation:
x = 1000^1 / (10^1 + 90)
x = 1000 / (10 + 90)
x = 1000 / 100
x = 10
Therefore, according to the equation, after 10 days, 10 people have heard the rumor.