Answer :
To find the value of [tex]\( f(5.5) \)[/tex] for the exponential function [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( f(3) = 4 \)[/tex] and [tex]\( f(8.5) = 76 \)[/tex], follow these steps:
1. Set up equations for the given points:
- From [tex]\( f(3) = 4 \)[/tex], we have [tex]\( a \cdot b^3 = 4 \)[/tex].
- From [tex]\( f(8.5) = 76 \)[/tex], we have [tex]\( a \cdot b^{8.5} = 76 \)[/tex].
2. Solve for [tex]\( b \)[/tex]:
- Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{8.5}}{a \cdot b^3} = \frac{76}{4}
\][/tex]
- Simplify the left side:
[tex]\[
b^{8.5 - 3} = 19
\][/tex]
[tex]\[
b^{5.5} = 19
\][/tex]
- Solve for [tex]\( b \)[/tex] by taking the fifth-and-a-half root of 19:
[tex]\[
b = 19^{\frac{1}{5.5}} \approx 1.708
\][/tex]
3. Solve for [tex]\( a \)[/tex]:
- Substitute [tex]\( b \)[/tex] back into the first equation:
[tex]\[
a \cdot (1.708)^3 = 4
\][/tex]
- Calculate [tex]\( (1.708)^3 \)[/tex] and solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{4}{(1.708)^3} \approx 0.803
\][/tex]
4. Calculate [tex]\( f(5.5) \)[/tex]:
- Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(5.5) = a \cdot b^{5.5}
\][/tex]
- Substitute the values:
[tex]\[
f(5.5) = 0.803 \cdot (1.708)^{5.5} \approx 15.25
\][/tex]
Therefore, the value of [tex]\( f(5.5) \)[/tex] is approximately [tex]\( 15.25 \)[/tex].
1. Set up equations for the given points:
- From [tex]\( f(3) = 4 \)[/tex], we have [tex]\( a \cdot b^3 = 4 \)[/tex].
- From [tex]\( f(8.5) = 76 \)[/tex], we have [tex]\( a \cdot b^{8.5} = 76 \)[/tex].
2. Solve for [tex]\( b \)[/tex]:
- Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{8.5}}{a \cdot b^3} = \frac{76}{4}
\][/tex]
- Simplify the left side:
[tex]\[
b^{8.5 - 3} = 19
\][/tex]
[tex]\[
b^{5.5} = 19
\][/tex]
- Solve for [tex]\( b \)[/tex] by taking the fifth-and-a-half root of 19:
[tex]\[
b = 19^{\frac{1}{5.5}} \approx 1.708
\][/tex]
3. Solve for [tex]\( a \)[/tex]:
- Substitute [tex]\( b \)[/tex] back into the first equation:
[tex]\[
a \cdot (1.708)^3 = 4
\][/tex]
- Calculate [tex]\( (1.708)^3 \)[/tex] and solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{4}{(1.708)^3} \approx 0.803
\][/tex]
4. Calculate [tex]\( f(5.5) \)[/tex]:
- Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(5.5) = a \cdot b^{5.5}
\][/tex]
- Substitute the values:
[tex]\[
f(5.5) = 0.803 \cdot (1.708)^{5.5} \approx 15.25
\][/tex]
Therefore, the value of [tex]\( f(5.5) \)[/tex] is approximately [tex]\( 15.25 \)[/tex].
