To determine the value of [tex]\( b \)[/tex] for the given exponential function in the form [tex]\( y = 20b^x \)[/tex] that passes through the points [tex]\((0, 20)\)[/tex] and [tex]\((1, 10)\)[/tex], follow these steps:
1. Start with the general form of the exponential equation:
[tex]\[
y = 20b^x
\][/tex]
2. Substitute the coordinates of the first point [tex]\((0, 20)\)[/tex] into the equation:
[tex]\[
20 = 20b^0
\][/tex]
Since any number to the power of 0 is 1, this simplifies to:
[tex]\[
20 = 20 \cdot 1
\][/tex]
3. This confirms that [tex]\( y = 20 \)[/tex] when [tex]\( x = 0 \)[/tex], which aligns with the given function. Hence, the equation remains:
[tex]\[
y = 20b^x
\][/tex]
4. Next, use the second point [tex]\((1, 10)\)[/tex] and substitute it into the equation:
[tex]\[
10 = 20b^1
\][/tex]
Simplify the equation:
[tex]\[
10 = 20b
\][/tex]
5. Solve for [tex]\( b \)[/tex] by dividing both sides of the equation by 20:
[tex]\[
b = \frac{10}{20}
\][/tex]
Simplify the fraction:
[tex]\[
b = 0.5
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] is [tex]\( 0.5 \)[/tex].
