Question

genesis is standing in a stadium and has the option of running up a staircase or down a staircase for exercise. the function ( f(x)=\frac{1}{25}(|3 x|+2 x) ) will determine how many calories genesis will burn if she runs up ( x ) steps. use a negative value for ( x ) if she runs down the stairs. find and interpret the given function values and determine an appropriate domain for the function.

answer: attempt 2 out of 2
( f(-25)= ) ( square ), means if genesis runs down ( vee 25 ) stairs, she will burn ( square ) calories. this interpretation does not make sense ( vee ) in the context of the problem.
( f(29)= ) ( square ), means if genesis runs up ( vee 29 ) stairs, she will burn ( square ) calories. this interpretation makes sense ( vee ) in the context of the problem.
( f(23.5)= ) ( square ), means if genesis runs up ( vee 23.5 ) stairs, she will burn ( square ) calories. this interpretation makes sense ( vee ) in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is all real numbers ( vee ).

Explanation:

Step1: Analyze the function \( f(x)=\frac{1}{25}(|3x| + 2x) \)

When \( x\geq0 \), \( |3x| = 3x \), so \( f(x)=\frac{1}{25}(3x + 2x)=\frac{1}{25}(5x)=\frac{x}{5} \). When \( x<0 \), \( |3x|=-3x \), so \( f(x)=\frac{1}{25}(-3x + 2x)=\frac{1}{25}(-x)=-\frac{x}{25} \).

Step2: Calculate \( f(29) \)

Since \( 29>0 \), use \( f(x)=\frac{x}{5} \). Substitute \( x = 29 \) into \( f(x)=\frac{x}{5} \), we get \( f(29)=\frac{29}{5}=5.8 \). This makes sense in the context of burning calories when running up stairs (positive number of stairs).

Step3: Calculate \( f(23.5) \)

Since \( 23.5>0 \), use \( f(x)=\frac{x}{5} \). Substitute \( x = 23.5 \) into \( f(x)=\frac{x}{5} \), we get \( f(23.5)=\frac{23.5}{5}=4.7 \). This makes sense in the context of burning calories when running up stairs (positive number of stairs).

Step4: Calculate \( f(-25) \)

Since \( -25<0 \), use \( f(x)=-\frac{x}{25} \). Substitute \( x=-25 \) into \( f(x)=-\frac{x}{25} \), we get \( f(-25)=-\frac{-25}{25}=1 \). But in the context of burning calories, running down stairs is a different physical activity. While the mathematical calculation is correct (\( f(-25) = 1 \)), the interpretation of burning 1 calorie when running down 25 stairs may not align with the typical understanding of calorie - burning mechanisms (usually more complex than just this simple formula for descending, as there is less work against gravity compared to ascending in a simple - model sense).

Answer:

For \( f(29) = 5.8 \), it means if Genesis runs up 29 stairs, she will burn 5.8 calories (makes sense in the context). For \( f(23.5)=4.7 \), it means if Genesis runs up 23.5 stairs, she will burn 4.7 calories (makes sense in the context). For \( f(-25) = 1 \), it means if Genesis runs down 25 stairs, she will burn 1 calorie (does NOT make sense in the context).