Question

side view of a waterslide
la tasa media de cambio de x = 0 a x = 15 se trata de −4.667¿cómo varía la tasa media de cambio de x = 0 a x = 20 ¿comparar con este numero?
la tasa media de cambio de x = 0 a x = 20 está disminuyendo
la tasa media de cambio de x = 0 a x = 15.
supongamos que la altura del tobogán es 2 pies, cuando x = 20¿cuál es la tasa de cambio promedio en toda la diapositiva?

Explanation:

First Part (Comparing Average Rates of Change)

Step1: Recall Average Rate of Change Formula

The average rate of change of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is \( \frac{f(b)-f(a)}{b - a} \).

Step2: Analyze \( x = 0 \) to \( x = 15 \)

At \( x = 0 \), height \( f(0)=80 \) ft. We need to find \( f(15) \). From the graph, at \( x = 15 \), let's assume the height (we can infer from the trend: at \( x = 10 \), height is 20? Wait, no, at \( x = 5 \) (first grid after 0) is 40, \( x = 10 \) is 20, \( x = 15 \) would be, let's see the pattern. Wait, the first part: from \( x = 0 \) to \( x = 5 \): \( \frac{40 - 80}{5 - 0}=\frac{-40}{5}=-8 \); \( x = 0 \) to \( x = 10 \): \( \frac{20 - 80}{10 - 0}=\frac{-60}{10}=-6 \); \( x = 0 \) to \( x = 15 \): let's say at \( x = 15 \), height is, maybe 10? Wait, the problem says the average rate from \( x = 0 \) to \( x = 15 \) is -4.667. Let's check: \( \frac{f(15)-80}{15 - 0}=-4.667 \), so \( f(15)=80+15\times(-4.667)\approx80 - 70 = 10 \) (approx).

Step3: Analyze \( x = 0 \) to \( x = 20 \)

At \( x = 20 \), height is 2 ft (given later). So average rate from \( x = 0 \) to \( x = 20 \) is \( \frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9 \)? Wait, no, wait the first part: the average rate from \( x = 0 \) to \( x = 15 \) is -4.667 (which is \( -\frac{70}{15}\approx - 4.666... \)). Now, from \( x = 0 \) to \( x = 20 \), with \( f(20)=2 \), the average rate is \( \frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9 \). Wait, but -3.9 is greater than -4.667? Wait, no, maybe I misread the graph. Wait the graph: at \( x = 0 \), height 80; \( x = 5 \), 40; \( x = 10 \), 20; \( x = 15 \), 10? \( x = 20 \), 0? But the problem says at \( x = 20 \), height is 2 ft. Wait, maybe the first part: the average rate from \( x = 0 \) to \( x = 15 \) is \( \frac{f(15)-80}{15} \). If at \( x = 15 \), the height is, say, 10 (from the graph's trend: 80, 40, 20, 10, 0...), then \( \frac{10 - 80}{15}=\frac{-70}{15}\approx - 4.666... \). Then at \( x = 20 \), height is 2, so \( \frac{2 - 80}{20}=\frac{-78}{20}=-3.9 \). Wait, but -3.9 is greater than -4.667 (since -3.9 is to the right of -4.667 on the number line, meaning it's larger). But the question is "La tasa media de cambio de \( x = 0 \) a \( x = 20 \) está disminuyendo la tasa media de cambio de \( x = 0 \) a \( x = 15 \)." Wait, no, the wording: "está disminuyendo" or "está aumentando"? Wait, maybe I made a mistake. Wait, the average rate of change is the slope. From \( x = 0 \) to \( x = 15 \), slope is -4.667. From \( x = 0 \) to \( x = 20 \), with \( f(20)=2 \), slope is \( \frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9 \). Since -3.9 > -4.667, the average rate of change from 0 to 20 is greater (less negative) than from 0 to 15. But the dropdown is "disminuyendo" (decreasing) or "aumentando"? Wait, maybe the graph's actual points: let's check the graph again. The graph has points: (0,80), (5,40), (10,20), (15,10), (20,0)? But the problem says at \( x = 20 \), height is 2. Maybe the problem's given \( x = 20 \) height is 2, so we use that. So average rate from 0 to 15: \( \frac{f(15)-80}{15}=-4.667 \), so \( f(15)=80 + 15\times(-4.667)\approx80 - 70 = 10 \). Then from 0 to 20: \( \frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9 \). Now, -3.9 is greater than -4.667 (because -3.9 is closer to zero). So the average rate of change from 0 to 20 is greater (less negative) than from 0 to 15. But the question is "La tasa media de cambio de \( x = 0 \) a \( x = 20 \) está disminuyendo la tasa media de cambio de \( x = 0 \) a \( x = 15 \)." Wait, maybe the wording is "está aumentando" or "está disminuyendo". Wait, may…

Step1: Identify Initial and Final Points

At \( x = 0 \), height \( f(0)=80 \) ft. At \( x = 20 \), height \( f(20)=2 \) ft.

Step2: Apply Average Rate of Change Formula

The average rate of change from \( x = 0 \) to \( x = 20 \) is \( \frac{f(20)-f(0)}{20 - 0} \).

Step3: Calculate the Value

Substitute \( f(20)=2 \) and \( f(0)=80 \):
\[
\frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9
\]
Or, if we simplify \( \frac{-78}{20}=-3.9 \) or \( -\frac{39}{10}=-3.9 \).

First Part Answer (Dropdown)

Assuming the correct comparison: since \( \text{ARC}_2=-3.9 \) and \( \text{ARC}_1=-4.667 \), \( \text{ARC}_2 > \text{ARC}_1 \), so the average rate of change from 0 to 20 is aumentando (increasing) compared to 0 to 15. But maybe the problem's graph has \( x = 20 \) as 0, then \( \text{ARC}_2=-4 \), still greater than -4.667. So the dropdown should be "aumentando" (but the user's text has "disminuyendo" as the current option, maybe a translation issue. Wait, "disminuyendo" means decreasing, "aumentando" means increasing. Since -3.9 > -4.667, the rate is increasing (becoming less negative), so the correct word is "aumentando".

Second Part Answer

The average rate of change is \( \frac{2 - 80}{20}=-3.9 \) (or \( -3.9 \) ft per ft, or \( -\frac{39}{10} \), or -3.9).

First Part (Dropdown) Answer:

aumentando (assuming the comparison: since -3.9 > -4.667, the rate is increasing)

Second Part (Average Rate of Change) Answer:

-3.9 (or \( -\frac{39}{10} \), or -3.9)

Answer:

Step1: Identify Initial and Final Points

At \( x = 0 \), height \( f(0)=80 \) ft. At \( x = 20 \), height \( f(20)=2 \) ft.

Step2: Apply Average Rate of Change Formula

The average rate of change from \( x = 0 \) to \( x = 20 \) is \( \frac{f(20)-f(0)}{20 - 0} \).

Step3: Calculate the Value

Substitute \( f(20)=2 \) and \( f(0)=80 \):
\[
\frac{2 - 80}{20 - 0}=\frac{-78}{20}=-3.9
\]
Or, if we simplify \( \frac{-78}{20}=-3.9 \) or \( -\frac{39}{10}=-3.9 \).

First Part Answer (Dropdown)

Assuming the correct comparison: since \( \text{ARC}_2=-3.9 \) and \( \text{ARC}_1=-4.667 \), \( \text{ARC}_2 > \text{ARC}_1 \), so the average rate of change from 0 to 20 is aumentando (increasing) compared to 0 to 15. But maybe the problem's graph has \( x = 20 \) as 0, then \( \text{ARC}_2=-4 \), still greater than -4.667. So the dropdown should be "aumentando" (but the user's text has "disminuyendo" as the current option, maybe a translation issue. Wait, "disminuyendo" means decreasing, "aumentando" means increasing. Since -3.9 > -4.667, the rate is increasing (becoming less negative), so the correct word is "aumentando".

Second Part Answer

The average rate of change is \( \frac{2 - 80}{20}=-3.9 \) (or \( -3.9 \) ft per ft, or \( -\frac{39}{10} \), or -3.9).

First Part (Dropdown) Answer:

aumentando (assuming the comparison: since -3.9 > -4.667, the rate is increasing)

Second Part (Average Rate of Change) Answer:

-3.9 (or \( -\frac{39}{10} \), or -3.9)