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Answer :
To solve for the value of [tex]\( f(0) \)[/tex] in the exponential function [tex]\( y = a b^x \)[/tex], you need to find the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. We're given two points: [tex]\( f(-3.5) = 10 \)[/tex] and [tex]\( f(0.5) = 89 \)[/tex]. Using these points, we'll set up and solve two equations.
### Step-by-step Solution
1. Set up the equations:
- For [tex]\( f(-3.5) = 10 \)[/tex], the equation becomes:
[tex]\[
10 = a \cdot b^{-3.5}
\][/tex]
- For [tex]\( f(0.5) = 89 \)[/tex], the equation is:
[tex]\[
89 = a \cdot b^{0.5}
\][/tex]
2. Solve the equations:
We need to solve for [tex]\( b \)[/tex] first. To do this, we divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{89}{10} = \frac{a \cdot b^{0.5}}{a \cdot b^{-3.5}}
\][/tex]
Simplifying the right side, we get:
[tex]\[
\frac{89}{10} = b^{0.5 - (-3.5)} = b^4
\][/tex]
So, to find [tex]\( b \)[/tex]:
[tex]\[
b = \left( \frac{89}{10} \right)^{1/4} \approx 1.727
\][/tex]
3. Find [tex]\( a \)[/tex]:
Now that we have [tex]\( b \)[/tex], we can substitute back into one of the original equations to find [tex]\( a \)[/tex]. Using the first equation:
[tex]\[
10 = a \cdot (1.727)^{-3.5}
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 10 \cdot (1.727)^{3.5} \approx 67.72
\][/tex]
4. Calculate [tex]\( f(0) \)[/tex]:
Now, use [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = a \cdot b^0 = a
\][/tex]
Therefore, since [tex]\( f(0) = a \)[/tex], the value of [tex]\( f(0) \)[/tex] is approximately 67.72 when rounded to the nearest hundredth.
Thus, the value of [tex]\( f(0) \)[/tex] is [tex]\( \boxed{67.72} \)[/tex].
### Step-by-step Solution
1. Set up the equations:
- For [tex]\( f(-3.5) = 10 \)[/tex], the equation becomes:
[tex]\[
10 = a \cdot b^{-3.5}
\][/tex]
- For [tex]\( f(0.5) = 89 \)[/tex], the equation is:
[tex]\[
89 = a \cdot b^{0.5}
\][/tex]
2. Solve the equations:
We need to solve for [tex]\( b \)[/tex] first. To do this, we divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{89}{10} = \frac{a \cdot b^{0.5}}{a \cdot b^{-3.5}}
\][/tex]
Simplifying the right side, we get:
[tex]\[
\frac{89}{10} = b^{0.5 - (-3.5)} = b^4
\][/tex]
So, to find [tex]\( b \)[/tex]:
[tex]\[
b = \left( \frac{89}{10} \right)^{1/4} \approx 1.727
\][/tex]
3. Find [tex]\( a \)[/tex]:
Now that we have [tex]\( b \)[/tex], we can substitute back into one of the original equations to find [tex]\( a \)[/tex]. Using the first equation:
[tex]\[
10 = a \cdot (1.727)^{-3.5}
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 10 \cdot (1.727)^{3.5} \approx 67.72
\][/tex]
4. Calculate [tex]\( f(0) \)[/tex]:
Now, use [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = a \cdot b^0 = a
\][/tex]
Therefore, since [tex]\( f(0) = a \)[/tex], the value of [tex]\( f(0) \)[/tex] is approximately 67.72 when rounded to the nearest hundredth.
Thus, the value of [tex]\( f(0) \)[/tex] is [tex]\( \boxed{67.72} \)[/tex].
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