Math / Calculus

QuestionThe horsepower, $H(s)$ , required for a racecar to overcome wind resistance is given by the function $H(s)=0.003 s^{2}+0.07 s-0.027$ , where $s$ is the speed of the car in miles per hour. What is the average rate of change in horsepower per unit speed if the racecar increases in speed from 80 mph to 100 mph ? 12.2 20.0 1.64 0.61

Studdy Solution

STEP 1

What is this asking? How much extra horsepower is needed on average for each additional mph between 80 mph and 100 mph? Watch out! Don't mix up speed and horsepower!
Horsepower depends on speed.

STEP 2

1. Calculate Horsepower at 80 mph
2. Calculate Horsepower at 100 mph
3. Calculate the Average Rate of Change

STEP 3

Let's plug in our initial speed, $s = 80$ mph, into our horsepower formula: $H(s) = 0.003s^2 + 0.07s - 0.027$ .

STEP 4

$H(80) = 0.003 \cdot (80)^2 + 0.07 \cdot 80 - 0.027$

STEP 5

$H(80) = 0.003 \cdot 6400 + 5.6 - 0.027$

STEP 6

$H(80) = 19.2 + 5.6 - 0.027$

STEP 7

$H(80) = 24.773$ So, at 80 mph, we need 24.773 horsepower!

STEP 8

Now, let's zoom zoom to $s = 100$ mph and plug it in:

STEP 9

$H(100) = 0.003 \cdot (100)^2 + 0.07 \cdot 100 - 0.027$

STEP 10

$H(100) = 0.003 \cdot 10000 + 7 - 0.027$

STEP 11

$H(100) = 30 + 7 - 0.027$

STEP 12

$H(100) = 36.973$ Wow, we need 36.973 horsepower at 100 mph!
That's a big jump!

STEP 13

The average rate of change tells us the average increase in horsepower for each mph increase in speed.
It's like finding the slope between two points!

STEP 14

Our change in horsepower is $36.973 - 24.773 = 12.2$ .

STEP 15

Our change in speed is $100 - 80 = 20$ .

STEP 16

Average Rate of Change $= \frac{\text{Change in Horsepower}}{\text{Change in Speed}}$ .

STEP 17

$\frac{12.2}{20} = 0.61$ So, on average, we need 0.61 more horsepower for each additional mph between 80 mph and 100 mph.

STEP 18

The average rate of change in horsepower is 0.61!

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Studdy Solution

STEP 1

What is this asking? How much *extra* horsepower is needed *on average* for each additional mph between 80 mph and 100 mph? Watch out! Don't mix up speed and horsepower!Horsepower *depends* on speed.

STEP 2

1. Calculate Horsepower at 80 mph2. Calculate Horsepower at 100 mph3. Calculate the Average Rate of Change

STEP 3

Let's **plug in** our **initial speed**, s=80s = 80s=80 mph, into our horsepower formula: H(s)=0.003s2+0.07s−0.027H(s) = 0.003s^2 + 0.07s - 0.027H(s)=0.003s2+0.07s−0.027.

STEP 4

H(80)=0.003⋅(80)2+0.07⋅80−0.027H(80) = 0.003 \cdot (80)^2 + 0.07 \cdot 80 - 0.027H(80)=0.003⋅(80)2+0.07⋅80−0.027

STEP 5

H(80)=0.003⋅6400+5.6−0.027H(80) = 0.003 \cdot 6400 + 5.6 - 0.027H(80)=0.003⋅6400+5.6−0.027

STEP 6

H(80)=19.2+5.6−0.027H(80) = 19.2 + 5.6 - 0.027H(80)=19.2+5.6−0.027

STEP 7

H(80)=24.773H(80) = 24.773H(80)=24.773 So, at 80 mph, we need **24.773 horsepower**!

STEP 8

Now, let's **zoom zoom** to s=100s = 100s=100 mph and **plug it in**:

STEP 9

H(100)=0.003⋅(100)2+0.07⋅100−0.027H(100) = 0.003 \cdot (100)^2 + 0.07 \cdot 100 - 0.027H(100)=0.003⋅(100)2+0.07⋅100−0.027

STEP 10

H(100)=0.003⋅10000+7−0.027H(100) = 0.003 \cdot 10000 + 7 - 0.027H(100)=0.003⋅10000+7−0.027

STEP 11

H(100)=30+7−0.027H(100) = 30 + 7 - 0.027H(100)=30+7−0.027

STEP 12

H(100)=36.973H(100) = 36.973H(100)=36.973 Wow, we need **36.973 horsepower** at 100 mph!That's a *big* jump!

STEP 13

The **average rate of change** tells us the *average* increase in horsepower for each mph increase in speed.It's like finding the slope between two points!

STEP 14

Our **change in horsepower** is 36.973−24.773=12.236.973 - 24.773 = 12.236.973−24.773=12.2.

STEP 15

Our **change in speed** is 100−80=20100 - 80 = 20100−80=20.

STEP 16

**Average Rate of Change** =Change in HorsepowerChange in Speed= \frac{\text{Change in Horsepower}}{\text{Change in Speed}}=Change in SpeedChange in Horsepower​.

STEP 17

12.220=0.61\frac{12.2}{20} = 0.612012.2​=0.61 So, on average, we need **0.61 more horsepower** for each additional mph between 80 mph and 100 mph.

STEP 18

The average rate of change in horsepower is **0.61**!

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What is this asking? How much extra horsepower is needed on average for each additional mph between 80 mph and 100 mph? Watch out! Don't mix up speed and horsepower!
Horsepower depends on speed.

1. Calculate Horsepower at 80 mph
2. Calculate Horsepower at 100 mph
3. Calculate the Average Rate of Change

Let's plug in our initial speed, $s = 80$ mph, into our horsepower formula: $H(s) = 0.003s^2 + 0.07s - 0.027$ .

$H(80) = 0.003 \cdot (80)^2 + 0.07 \cdot 80 - 0.027$

$H(80) = 0.003 \cdot 6400 + 5.6 - 0.027$

$H(80) = 19.2 + 5.6 - 0.027$

$H(80) = 24.773$ So, at 80 mph, we need 24.773 horsepower!

Now, let's zoom zoom to $s = 100$ mph and plug it in:

$H(100) = 0.003 \cdot (100)^2 + 0.07 \cdot 100 - 0.027$

$H(100) = 0.003 \cdot 10000 + 7 - 0.027$

$H(100) = 30 + 7 - 0.027$

$H(100) = 36.973$ Wow, we need 36.973 horsepower at 100 mph!
That's a big jump!

The average rate of change tells us the average increase in horsepower for each mph increase in speed.
It's like finding the slope between two points!

Our change in horsepower is $36.973 - 24.773 = 12.2$ .

Our change in speed is $100 - 80 = 20$ .

Average Rate of Change $= \frac{\text{Change in Horsepower}}{\text{Change in Speed}}$ .

$\frac{12.2}{20} = 0.61$ So, on average, we need 0.61 more horsepower for each additional mph between 80 mph and 100 mph.

The average rate of change in horsepower is 0.61!