The rate of auto thefts triples every 6 months.
(a) Determine, to two decimal places, the base b for an exponential model y = Ab^t of the rate of auto thefts as a function of time in months.
b =
(b) Find the doubling time to the nearest tenth of a month.
months
(a)
y = A*3^(t/6) = A*(3^(1/6))^t = A*1.20^t
(b)
solve 1.20^t = 2
for t
To determine the base, b, for an exponential model y = Ab^t, where y is the rate of auto thefts and t is the time in months, we can use the information that the rate of auto thefts triples every 6 months.
(a) Since the rate triples every 6 months, we can express this relationship as:
(1) 3 = b^(6)
where b represents the base.
To find the value of b, we need to solve for it in equation (1). Taking the natural logarithm (ln) of both sides:
ln(3) = ln(b^(6))
Using the logarithm property of exponents, we can bring the exponent down:
ln(3) = 6 * ln(b)
Now, divide both sides by 6 to isolate ln(b):
ln(b) = ln(3) / 6
To find the value of b, we need to take the exponential (e) of both sides:
b = e^(ln(3) / 6)
Using a calculator, we can evaluate this expression to two decimal places:
b ≈ 1.115
Therefore, the base, b, for the exponential model is approximately 1.115.
(b) To find the doubling time, we need to determine the time it takes for the rate of auto thefts to double. In an exponential model, the doubling time is given by the formula:
Doubling time = ln(2) / ln(b)
Using the value of b we found in part (a), we can calculate the doubling time:
Doubling time ≈ ln(2) / ln(1.115)
Using a calculator, we can evaluate this expression to the nearest tenth of a month:
Doubling time ≈ 6.23 months
Therefore, the doubling time is approximately 6.2 months.
(a) To determine the base b for an exponential model y = Ab^t, we need to find the value of b that will result in tripling the rate every 6 months.
Let's assume the initial rate is y₀ and the rate after 6 months is y₁, then according to the information provided:
y₁ = 3y₀
Since we have an exponential model, we can express y₀ and y₁ as:
y₀ = Ab^0
y₁ = Ab^6
Substituting these values into the equation 3y₀ = y₁, we get:
3Ab^0 = Ab^6
Since b^0 equals 1, we can simplify the equation to:
3A = Ab^6
Dividing both sides by A, we get:
3 = b^6
Taking the sixth root of both sides:
b = ∛3 ≈ 1.442
Therefore, the base (b) for the exponential model is approximately 1.442.
(b) To find the doubling time, we need to determine the time it takes for the rate to double. Since b represents the rate at which the model grows, we can find the doubling time by solving the equation:
2 = b^t
Taking the logarithm base b of both sides:
logb(2) = logb(b^t)
Using the logarithm property logb(b^a) = a, we get:
logb(2) = t
Substituting the value of b we found earlier:
log1.442(2) = t
Using a calculator, we find:
t ≈ 0.4803
Therefore, the doubling time is approximately 0.5 months.