A population of bacteria is growing according to the equation P(t)=850e^0.15t. Estimate when the population will exceed 2253.
t=
To estimate when the population will exceed 2253, we need to find the value of t in the equation P(t) = 2253.
The given equation is P(t) = 850e^(0.15t), where P(t) represents the population at time t.
So, we substitute P(t) = 2253 into the equation and solve for t:
2253 = 850e^(0.15t)
To solve for t, we need to isolate the exponential term e^(0.15t).
Divide both sides of the equation by 850:
2253/850 = e^(0.15t)
Simplify the left side:
2.65 = e^(0.15t)
To solve for t, we need to take the natural logarithm of both sides:
ln(2.65) = ln(e^(0.15t))
Using the property of logarithms, the exponent can be brought down as a coefficient:
ln(2.65) = 0.15t ln(e)
Since ln(e) is equal to 1, we can simplify the equation:
ln(2.65) = 0.15t
Now, divide both sides of the equation by 0.15:
ln(2.65)/0.15 = t
Using a calculator, evaluate ln(2.65)/0.15 to find the value of t.
Hence, the estimated value of t when the population exceeds 2253 is the result of ln(2.65)/0.15.
850 e^0.15t > 2253
e^0.15t > 2.65058
0.15t > ln 2.65058
t > 6.5