Question

question 9 of 41
the volume of the prism is given by the function ( f(x) = \frac{x^3}{2} ), where ( x ) = the side length in meters of the base, and ( f(x) ) = the prisms volume in cubic meters.
which is a true statement about the rate of change of ( f(x) ) as ( x ) increases?

• between ( x = 2 ) and ( x = 4 ), the volume increases by an average of 28 cubic meters for each 1 - meter increase in ( x ).
• between ( x = 1 ) and ( x = 3 ), the volume increases by an average of 13 cubic meters for each 1 - meter increase in ( x ).
• between ( x = 2 ) and ( x = 4 ), the volume increases by an average of 14 cubic meters for each 1 - meter increase in ( x ).
• between ( x = 1 ) and ( x = 3 ), the volume increases by an average of 27 cubic meters for each 1 - meter increase in ( x ).

Explanation:

Step1: Recall Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( f(x)=\frac{x^3}{2} \), so we'll calculate this for each interval.

Step2: Analyze Option 1 (x=2 to x=4)

First, find \( f(2) \) and \( f(4) \).
\( f(2)=\frac{2^3}{2}=\frac{8}{2} = 4 \)
\( f(4)=\frac{4^3}{2}=\frac{64}{2}=32 \)
Average rate of change: \(\frac{32 - 4}{4 - 2}=\frac{28}{2}=14\). Wait, but the option says 28 per 1 - meter. No, wait, \( 4 - 2 = 2 \) meters interval. Wait, no, the option says "for each 1 - meter increase". Wait, maybe I misread. Wait, the interval length is \( 4 - 2 = 2 \), so average rate per 1 - meter would be \(\frac{28}{2}=14\)? Wait no, wait the formula is \(\frac{f(b)-f(a)}{b - a}\), which is average rate over the interval. So for x=2 to x=4: \( f(4)-f(2)=32 - 4 = 28 \), interval length \( 4 - 2 = 2 \), so average rate is \(\frac{28}{2}=14\) per meter? Wait no, the option says "the volume increases by an average of 28 cubic meters for each 1 - meter increase in x". But \(\frac{28}{2}=14\), so this is wrong.

Step3: Analyze Option 2 (x=1 to x=3)

\( f(1)=\frac{1^3}{2}=0.5 \), \( f(3)=\frac{27}{2}=13.5 \)
Change: \( 13.5 - 0.5 = 13 \), interval length \( 3 - 1 = 2 \), average rate: \(\frac{13}{2}=6.5\). Not 13 per meter. So wrong.

Step4: Analyze Option 3 (x=2 to x=4)

Wait, earlier for x=2 to x=4: \( f(4)-f(2)=32 - 4 = 28 \), interval length \( 4 - 2 = 2 \), average rate \(\frac{28}{2}=14\) per meter? Wait no, the option says "between x=2 and x=4, the volume increases by an average of 14 cubic meters for each 1 - meter increase in x". Wait, \(\frac{28}{2}=14\), yes! Wait, wait, no: the interval from 2 to 4 is 2 meters. So the total change is 28, over 2 meters, so per meter is 14. Wait, but the option says "for each 1 - meter increase". So average rate is 14. Wait, but let's check the last option.

Step5: Analyze Option 4 (x=1 to x=3)

Wait, no, the last option: "between x=1 and x=3, the volume increases by an average of 27 cubic meters for each 1 - meter increase in x". \( f(3)-f(1)=13.5 - 0.5 = 13 \), interval length 2, so \(\frac{13}{2}=6.5\), no. Wait, wait the first option (the selected one in the image? Wait the image shows a radio button on the last option? Wait no, let's re - check the function: \( f(x)=\frac{x^3}{2} \).

Wait, let's check the option: "BETWEEN x = 1 AND x = 3, THE VOLUME INCREASES BY AN AVERAGE OF 27 CUBIC METERS FOR EACH 1 - METER INCREASE IN x". Wait \( f(3)=\frac{27}{2}=13.5 \), \( f(1)=\frac{1}{2}=0.5 \), difference is 13, over 2 meters, so 6.5 per meter. No.

Wait, wait maybe I made a mistake. Let's check the option: "BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN x". Let's recalculate: \( f(2)=\frac{8}{2}=4 \), \( f(4)=\frac{64}{2}=32 \), difference is 28, over 2 meters (from 2 to 4 is 2 meters), so 28 / 2 = 14 per meter. Yes! So that's correct. Wait, but the option is "BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN x". So that's the third option? Wait the options are:

1. BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 28 CUBIC METERS FOR EACH 1 - METER INCREASE IN x. (Wrong, 28 / 2 = 14)

2. BETWEEN x = 1 AND x = 3, THE VOLUME INCREASES BY AN AVERAGE OF 13 CUBIC METERS FOR EACH 1 - METER INCREASE IN x. (13 / 2 = 6.5, wrong)

3. BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN x. (28 / 2 = 14, correct)

4. BETWEEN x = 1 AND x = 3, THE VOLUME INCREASES BY AN AVERAGE OF 27 CUBIC METERS FOR EACH 1 - METER INCREASE IN x. (13 / 2 = 6.5, wrong)

Wait, so the correct option is the third one: "BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN x". Wait, but let's confirm:

Average rate of change formula: \(\frac{f(b)-f(a)}{b - a}\). For x=2 to x=4: \( f(4)-f(2)=32 - 4 = 28 \), \( b - a = 4 - 2 = 2 \), so \(\frac{28}{2}=14\). So the average rate is 14 cubic meters per meter increase in x. So the third option (the one that says between x=2 and x=4, average of 14 per 1 - meter) is correct.

Wait, but in the image, the selected option is the last one? Wait no, the image's radio button is on the last option, but that's wrong. Wait maybe I misread the function. Wait the function is \( f(x)=\frac{x^3}{2} \)? Wait no, maybe it's \( f(x)=x^3/2 \)? Wait let's re - check:

For x=1: \( 1^3/2 = 0.5 \)

x=2: \( 8/2 = 4 \)

x=3: \( 27/2 = 13.5 \)

x=4: \( 64/2 = 32 \)

So between x=2 and x=4: change is 32 - 4 = 28, over 2 meters, so 14 per meter. So the option "BETWEEN x = 2 AND x = 4, THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN x" is correct.

Answer:

The correct option is: BETWEEN \( x = 2 \) AND \( x = 4 \), THE VOLUME INCREASES BY AN AVERAGE OF 14 CUBIC METERS FOR EACH 1 - METER INCREASE IN \( x \) (the third option in the list, assuming the options are ordered as: 1. x=2 - 4, 28; 2. x=1 - 3, 13; 3. x=2 - 4, 14; 4. x=1 - 3, 27)