Answer :
Sure! Let's solve the problem step-by-step.
We are given the function [tex]\( f(t) = D \cdot e^{rt} \)[/tex] and told that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].
To find the value of [tex]\( D \)[/tex], let's follow these steps:
1. Understand the Function: The function involved here is an exponential function where [tex]\( f(t) = D \cdot e^{rt} \)[/tex].
2. Substitute Known Values: We are given that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. Plug these into the equation:
[tex]\[
246.4 = D \cdot e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponential Part: First, compute the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
So, we need to calculate [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( D \)[/tex]: The equation becomes:
[tex]\[
246.4 = D \cdot e^{0.16}
\][/tex]
To isolate [tex]\( D \)[/tex], divide both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
D = \frac{246.4}{e^{0.16}}
\][/tex]
5. Estimate [tex]\( D \)[/tex]: Once you compute [tex]\( e^{0.16} \)[/tex], use this to find the approximate value of [tex]\( D \)[/tex].
When you compute this, you'll find that [tex]\( D \approx 210 \)[/tex].
correct answer to the problem is (B) 210.
We are given the function [tex]\( f(t) = D \cdot e^{rt} \)[/tex] and told that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].
To find the value of [tex]\( D \)[/tex], let's follow these steps:
1. Understand the Function: The function involved here is an exponential function where [tex]\( f(t) = D \cdot e^{rt} \)[/tex].
2. Substitute Known Values: We are given that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. Plug these into the equation:
[tex]\[
246.4 = D \cdot e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponential Part: First, compute the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
So, we need to calculate [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( D \)[/tex]: The equation becomes:
[tex]\[
246.4 = D \cdot e^{0.16}
\][/tex]
To isolate [tex]\( D \)[/tex], divide both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
D = \frac{246.4}{e^{0.16}}
\][/tex]
5. Estimate [tex]\( D \)[/tex]: Once you compute [tex]\( e^{0.16} \)[/tex], use this to find the approximate value of [tex]\( D \)[/tex].
When you compute this, you'll find that [tex]\( D \approx 210 \)[/tex].
correct answer to the problem is (B) 210.
