If $f(x)$ is an exponential function where $f(-1.5)=26$ and $f(5.5)=7$ , then find the value of $f(10)$ , to the nearest hundredth.

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Q. If

f(x)

is an exponential function where

f(-1.5)=26

and

f(5.5)=7

, then find the value of

f(10)

, to the nearest hundredth.

Given Points and Function Form: We are given two points on the exponential function: $(-1.5, 26)$ and $(5.5, 7)$ . We can use these points to find the base of the exponential function. The general form of an exponential function is $f(x) = a \cdot b^x$ , where $a$ is the initial value and $b$ is the base.
Set Up Equations: First, we need to set up two equations using the given points and the general form of the exponential function: $\newline$ $1$ ) $26 = a \cdot b^{-1.5}$ $\newline$ $2$ ) $7 = a \cdot b^{5.5}$
Solve for Base: We can solve these equations simultaneously to find the values of $a$ and $b$ . Let's divide the second equation by the first equation to eliminate the variable $a$ : $\left(\frac{7}{26}\right) = \frac{a \cdot b^{5.5}}{a \cdot b^{-1.5}}$
Calculate Value of b: Simplifying the right side of the equation, we get: $\newline$ $(\frac{7}{26}) = b^{(5.5 + 1.5)}$ $\newline$ $(\frac{7}{26}) = b^7$
Find Value of a: Now we can find the value of $b$ by taking the $7$ th root of both sides: $\newline$ $b = \left(\frac{7}{26}\right)^{\frac{1}{7}}$
Write Exponential Function: Calculating the value of $b$ : $b \approx (0.26923)^{\frac{1}{7}}$ $b \approx 0.76923$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$ Substitute the value of $b$ into the equation: $\newline$ $26 = a \cdot (0.76923)^{-1.5}$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$ Substitute the value of $b$ into the equation: $\newline$ $26 = a \cdot (0.76923)^{-1.5}$ Solve for $a$ : $\newline$ $a \approx 26 / (0.76923)^{-1.5}$ $\newline$ $a \approx 26 / 2.21868$ $\newline$ $a \approx 11.717$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$ Substitute the value of $b$ into the equation: $\newline$ $26 = a \cdot (0.76923)^{-1.5}$ Solve for $a$ : $\newline$ $a \approx 26 / (0.76923)^{-1.5}$ $\newline$ $a \approx 26 / 2.21868$ $\newline$ $a \approx 11.717$ Now we have both $a$ and $b$ , and we can write the exponential function as: $\newline$ $b$ $2$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$ Substitute the value of $b$ into the equation: $\newline$ $26 = a \cdot (0.76923)^{-1.5}$ Solve for $a$ : $\newline$ $a \approx 26 / (0.76923)^{-1.5}$ $\newline$ $a \approx 26 / 2.21868$ $\newline$ $a \approx 11.717$ Now we have both $a$ and $b$ , and we can write the exponential function as: $\newline$ $b$ $2$ Finally, we can find $f(10)$ using the exponential function: $\newline$ $b$ $4$
Find $f(10)$ : Now that we have the value of $b$ , we can use one of the original equations to find the value of $a$ . Let's use the first equation: $\newline$ $26 = a \cdot b^{-1.5}$ Substitute the value of $b$ into the equation: $\newline$ $26 = a \cdot (0.76923)^{-1.5}$ Solve for $a$ : $\newline$ $a \approx 26 / (0.76923)^{-1.5}$ $\newline$ $a \approx 26 / 2.21868$ $\newline$ $a \approx 11.717$ Now we have both $a$ and $b$ , and we can write the exponential function as: $\newline$ $b$ $2$ Finally, we can find $f(10)$ using the exponential function: $\newline$ $b$ $4$ Calculating $f(10)$ : $\newline$ $b$ $6$ $\newline$ $b$ $7$