Answer :
To solve the problem, we want to find the value of [tex]\( f(1.5) \)[/tex] for the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex], given the conditions [tex]\( f(0.5) = 26 \)[/tex] and [tex]\( f(1) = 66 \)[/tex].
Let's follow these steps:
1. Initial Equations:
- Using the information given:
- [tex]\( f(0.5) = a \cdot b^{0.5} = 26 \)[/tex]
- [tex]\( f(1) = a \cdot b^1 = 66 \)[/tex]
2. Find the Value of [tex]\( b \)[/tex]:
- Divide the equation for [tex]\( f(1) \)[/tex] by the equation for [tex]\( f(0.5) \)[/tex]:
[tex]\[
\frac{a \cdot b^1}{a \cdot b^{0.5}} = \frac{66}{26}
\][/tex]
- Simplify the left side:
[tex]\[
b^{1 - 0.5} = b^{0.5} = \frac{66}{26}
\][/tex]
- Solve for [tex]\( b \)[/tex] by squaring both sides:
[tex]\[
b = \left( \frac{66}{26} \right)^2
\][/tex]
3. Find the Value of [tex]\( a \)[/tex]:
- Use the value of [tex]\( b \)[/tex] found in the first equation:
[tex]\[
a \cdot b^{0.5} = 26
\][/tex]
- Substitute the expression for [tex]\( b \)[/tex] and solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{26}{b^{0.5}} = \frac{26}{\frac{66}{26}}
\][/tex]
- Simplify:
[tex]\[
a = 26 \cdot \frac{26}{66}
\][/tex]
4. Calculate [tex]\( f(1.5) \)[/tex]:
- Use the function [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = a \cdot b^{1.5}
\][/tex]
- Insert the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(1.5) = \left(26 \cdot \frac{26}{66}\right) \cdot \left(\frac{66}{26}\right)^{1.5}
\][/tex]
5. Final Answer:
- Simplify and calculate:
[tex]\[
f(1.5) \approx 66.00
\][/tex]
- Therefore, the approximate value is [tex]\( f(1.5) = 66.00 \)[/tex].
This gives you the required value for [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth.
Let's follow these steps:
1. Initial Equations:
- Using the information given:
- [tex]\( f(0.5) = a \cdot b^{0.5} = 26 \)[/tex]
- [tex]\( f(1) = a \cdot b^1 = 66 \)[/tex]
2. Find the Value of [tex]\( b \)[/tex]:
- Divide the equation for [tex]\( f(1) \)[/tex] by the equation for [tex]\( f(0.5) \)[/tex]:
[tex]\[
\frac{a \cdot b^1}{a \cdot b^{0.5}} = \frac{66}{26}
\][/tex]
- Simplify the left side:
[tex]\[
b^{1 - 0.5} = b^{0.5} = \frac{66}{26}
\][/tex]
- Solve for [tex]\( b \)[/tex] by squaring both sides:
[tex]\[
b = \left( \frac{66}{26} \right)^2
\][/tex]
3. Find the Value of [tex]\( a \)[/tex]:
- Use the value of [tex]\( b \)[/tex] found in the first equation:
[tex]\[
a \cdot b^{0.5} = 26
\][/tex]
- Substitute the expression for [tex]\( b \)[/tex] and solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{26}{b^{0.5}} = \frac{26}{\frac{66}{26}}
\][/tex]
- Simplify:
[tex]\[
a = 26 \cdot \frac{26}{66}
\][/tex]
4. Calculate [tex]\( f(1.5) \)[/tex]:
- Use the function [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = a \cdot b^{1.5}
\][/tex]
- Insert the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(1.5) = \left(26 \cdot \frac{26}{66}\right) \cdot \left(\frac{66}{26}\right)^{1.5}
\][/tex]
5. Final Answer:
- Simplify and calculate:
[tex]\[
f(1.5) \approx 66.00
\][/tex]
- Therefore, the approximate value is [tex]\( f(1.5) = 66.00 \)[/tex].
This gives you the required value for [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth.
