Select the correct answer.
A geologist gathered data about the total shoreline and maximum depth of several area lakes and organized the data into this table.
| Total Shoreline (miles) | 22 | 17 | 10 | 23 | 12 | 35 | 7 |
| :---------------------- | :-- | :-- | :-- | :-- | :-- | :-- | :-- |
| Maximum Depth (feet) | 101 | 85 | 59 | 113 | 64 | 158 | 33 |
She then used a graphing tool to display the data in a scatter plot, with [tex]$x$[/tex] representing the total miles of shoreline and [tex]$y$[/tex] representing the maximum depth. She also used the graphing tool to find the equation of the line of best fit
[tex]$y=4.26 x+10.908$[/tex]
Based on the line of best fit, what is the approximate maximum depth of a lake that has 31 miles of shoreline?
A. 121 feet
B. 132 feet
C. 138 feet
D. 143 feet
Answer (2)
Substitute the given shoreline length ( x = 31 ) into the line of best fit equation: y = 4.26 x + 10.908 .
Calculate the result: y = 4.26 ( 31 ) + 10.908 = 132.06 + 10.908 = 142.968 .
Round the calculated depth to the nearest whole number to find the approximate depth.
The approximate maximum depth of the lake is 143 feet.
Explanation
Understanding the Problem We are given the equation of the line of best fit as y = 4.26 x + 10.908 , where x represents the total miles of shoreline and y represents the maximum depth in feet. We are asked to find the approximate maximum depth of a lake that has 31 miles of shoreline. This means we need to substitute x = 31 into the equation and solve for y .
Substitution Substitute x = 31 into the equation: y = 4.26 ( 31 ) + 10.908
Calculation Calculate the value of y :
y = 132.06 + 10.908 = 142.968 So, the approximate maximum depth is 142.968 feet.
Rounding and Conclusion Since we need to find the approximate maximum depth, we round the value of y to the nearest whole number. 142.968 rounded to the nearest whole number is 143. Therefore, the approximate maximum depth of a lake that has 31 miles of shoreline is 143 feet.
Examples
Understanding the relationship between shoreline length and maximum depth can help in lake management and conservation efforts. For example, if you're managing a lake and want to predict its maximum depth based on its shoreline, you can use this equation. This can be useful for stocking fish, managing aquatic vegetation, or predicting water quality parameters. This kind of analysis is also useful in environmental impact assessments where changes to shoreline may affect the lake's ecosystem.
To find the maximum depth for a lake with 31 miles of shoreline, we substitute x = 31 into the equation, which results in approximately 143 feet when rounded. Therefore, the best answer is 143 feet. Thus, the correct multiple-choice option is D. 143 feet.
;
