Pregunta
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If is an exponential function of the form of where and ,
then find the value of , to the nearest hundredth.

Ask by Valdez Powers. in the United States
Jan 23,2025

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Solución

Given that is an exponential function of the form , where and , we need to find the value of to the nearest hundredth.
To find the value of , we first need to determine the values of and using the given information.
From the given information, we have:
We can use these two equations to solve for and .
Let’s start by using the first equation to find the value of :
Now, let’s use the second equation to find the value of :
We can solve these two equations simultaneously to find the values of and .
Let’s solve these equations to find the values of and and then use them to find the value of .
Solve the system of equations by following steps:
  • step0: Solve using the substitution method:
  • step1: Calculate:
  • step2: Solve the equation:
  • step3: Substitute the value of
  • step4: Reduce the fraction:
  • step5: Swap the sides:
  • step6: Divide both sides:
  • step7: Divide the numbers:
  • step8: Raise both sides to the -th power
  • step9: Evaluate the power:
  • step10: Substitute the value of
  • step11: Simplify:
  • step12: Calculate:
  • step13: Check the solution:
  • step14: Rewrite:

    The solution to the system of equations is and .
Now that we have the values of and , we can find the value of using the exponential function .
Substitute the values of and into the function to find :
Let’s calculate the value of to the nearest hundredth.
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Evaluate the power:
  • step2: Rewrite the expression:
  • step3: Reduce the numbers:
  • step4: Multiply:
  • step5: Evaluate the power:

    The value of to the nearest hundredth is 182.25.

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Beyond the Answer

To solve for , we start with the given form of the function and the provided values and .
Setting up the equations with the known values:
  1. From :
  2. From :
Next, we can express in terms of :
Substituting this into equation (2):
Now, replacing from equation (1):
Solving for :
To find , we square both sides:
Now, substitute back into equation (1) to find :
Calculating :
Then,
Now we can express fully:
Finally, we need to find :
Calculating :
Then,
Thus, the value of to the nearest hundredth is:

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