Question
Shown is the graph of y=5^x.
Use the graph to identify the best estimate of 5^(2.75)
a. 75
b. 80
c. 39
d. 100
Use the graph to identify the best estimate of 5^(2.75)
a. 75
b. 80
c. 39
d. 100
Answers
GPT-4o mini
To estimate \( 5^{2.75} \) using the graph of \( y = 5^x \), we can analyze the behavior of the function.
We know that:
- \( 5^2 = 25 \)
- \( 5^3 = 125 \)
Since \( 2.75 \) is between \( 2 \) and \( 3 \), we can realize that \( 5^{2.75} \) will be between \( 25 \) and \( 125 \).
To narrow it down further, we can look for an approximate value.
Given that \( 5^{2.5} \) is approximately the square root of \( 5^5 \) (or \( 5^{2.5} \approx \sqrt{3125} \approx 55.9 \)), and we know \( 5^{2.75} \) will be a bit more than that (since \( 5^{2.75} \) is a bit larger than \( 5^{2.5} \) but less than \( 5^3 \)).
Thus:
- The potential options are \( 75, 80, 39, \) and \( 100 \).
- We can eliminate \( 39 \) right away because it's too low.
- Since \( 5^{2.75} \) will be less than \( 125 \) but definitely more than \( 75 \), we can reasonably guess that it should be closer to that range.
Looking at the two relevant and closer options:
- \( 75 \) is near, but lets see if \( 80 \) is a better fit.
Given these correlations, \( 5^{2.75} \) is best estimated at **80** based on the options provided.
So, the best estimated value of \( 5^{2.75} \) is:
**b. 80**
We know that:
- \( 5^2 = 25 \)
- \( 5^3 = 125 \)
Since \( 2.75 \) is between \( 2 \) and \( 3 \), we can realize that \( 5^{2.75} \) will be between \( 25 \) and \( 125 \).
To narrow it down further, we can look for an approximate value.
Given that \( 5^{2.5} \) is approximately the square root of \( 5^5 \) (or \( 5^{2.5} \approx \sqrt{3125} \approx 55.9 \)), and we know \( 5^{2.75} \) will be a bit more than that (since \( 5^{2.75} \) is a bit larger than \( 5^{2.5} \) but less than \( 5^3 \)).
Thus:
- The potential options are \( 75, 80, 39, \) and \( 100 \).
- We can eliminate \( 39 \) right away because it's too low.
- Since \( 5^{2.75} \) will be less than \( 125 \) but definitely more than \( 75 \), we can reasonably guess that it should be closer to that range.
Looking at the two relevant and closer options:
- \( 75 \) is near, but lets see if \( 80 \) is a better fit.
Given these correlations, \( 5^{2.75} \) is best estimated at **80** based on the options provided.
So, the best estimated value of \( 5^{2.75} \) is:
**b. 80**