Exponential functions word problem

In summary, exponential functions are mathematical functions in the form of y = ab^x, used to describe relationships where the output value changes at a constant rate with respect to the input value. They have various real-life applications such as modeling population growth and analyzing data sets. There is a difference between exponential growth and decay, and to solve an exponential function word problem, one must identify the given information and use algebraic techniques. Common misconceptions about exponential functions include thinking that they always lead to infinite values, only apply to large numbers, and have a constant rate of change.
  • #1
Coco12
272
0

Homework Statement



The function T=190(1/2)^1/10t can be used to determine the length of time t, in hrs that milk will remain fresh. T is the storage temp. In Celsius
How long will milk remain fresh at 22 degrees Celsius

Homework Equations



Bases have to be same then exponents will equal one another

The Attempt at a Solution


In this equation , we need to solve for the t in the exponent. I thought you has to divide 22 (T) by 190 and tried to make it so that the base is the same as (1/2) but the number when you divide 22 by 190 is a very long decimal and I don't know how to do that. So far we have not done Logs yet in class so how can I solve that without using logs?
 
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  • #2
Coco12 said:

Homework Statement



The function T=190(1/2)^1/10t can be used to determine the length of time t, in hrs that milk will remain fresh. T is the storage temp. In Celsius
How long will milk remain fresh at 22 degrees Celsius

Homework Equations



Bases have to be same then exponents will equal one another

The Attempt at a Solution


In this equation , we need to solve for the t in the exponent. I thought you has to divide 22 (T) by 190 and tried to make it so that the base is the same as (1/2) but the number when you divide 22 by 190 is a very long decimal and I don't know how to do that. So far we have not done Logs yet in class so how can I solve that without using logs?

"Bases have to be same then exponents will equal one another"

It's unclear what this statement means, and I don't think it will help solve the problem.

Don't you understand how logarithms work? Try taking the logarithm of both sides of the Temperature equation.

Remember that e^(ln x) = x
 
  • #3
Coco12 said:

Homework Statement



The function T=190(1/2)^1/10t can be used to determine the length of time t, in hrs that milk will remain fresh. T is the storage temp. In Celsius
How long will milk remain fresh at 22 degrees Celsius

Homework Equations



Bases have to be same then exponents will equal one another

The Attempt at a Solution


In this equation , we need to solve for the t in the exponent. I thought you has to divide 22 (T) by 190 and tried to make it so that the base is the same as (1/2) but the number when you divide 22 by 190 is a very long decimal and I don't know how to do that. So far we have not done Logs yet in class so how can I solve that without using logs?
Surely you are not going to let a number, even a very long one, scare you? The fact that you have not done logarithms is more important. 22/190= 0.115789, approximately, and if I don't want to (or can't) use logarithms then I had better think in terms of powers of 1/2. I know that [itex](1/2)^3= 1/8= 0.125[/itex] and [itex](1/2)^4= 1/16= 0.0625[/itex] so the nearest I can do is that 1/(10t) is about 3.
 
  • #4
yes, without using logarithms, the simplest method is probably the tried-and-tested, backup-option of just trying a few values, and choosing the one that fits the equation the closest.
 
  • #5
Ok thank you , could I also use a graphing calculator?
 
  • #6
Is the exponent (1/10)t or 1/(10t) ? Also you can press the little X2 button to do exponents properly.
 
  • #7
@Coco12: yeah, that should work. when it draws the graph of the function, it will essentially do the same thing you are (i.e. choose a bunch of values and calculate what the function evaluates to). except it will generally be faster than a person. be careful to avoid t=0
 
  • #8
BMW said:
Is the exponent (1/10)t or 1/(10t) ? Also you can press the little X2 button to do exponents properly.

To get to that extended menu, click Go Advanced. The X2 let's you enter exponents so they appear in a nice format.
 

What is an exponential function?

An exponential function is a mathematical function in the form of y = ab^x, where a and b are constants and x is the independent variable. It describes a relationship where the output value (y) changes at a constant rate with respect to the input value (x).

How are exponential functions used in real life?

Exponential functions can be used to model various real-life scenarios such as population growth, compound interest, and radioactive decay. They can also be used to analyze data sets that show exponential patterns, such as stock market trends or the spread of diseases.

What is the difference between exponential growth and decay?

Exponential growth occurs when the value of the independent variable (x) increases, resulting in an increasing value of the dependent variable (y). On the other hand, exponential decay occurs when the value of x decreases, resulting in a decreasing value of y.

How do you solve an exponential function word problem?

To solve an exponential function word problem, first identify the given information and what needs to be solved for. Then, substitute the known values into the function and use algebraic techniques to isolate the variable. Finally, solve for the variable using logarithms or other methods.

What are some common misconceptions about exponential functions?

One common misconception is that exponential growth always leads to infinite values, when in reality, many real-life scenarios have limitations or constraints that prevent this. Another misconception is that exponential functions only apply to large numbers, when in fact, they can also be used to model small values. Additionally, some people may think that all exponential functions have a constant rate of change, but this is not always the case as the base (b) can vary.

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