Math / Algebra

Question24. Question from 1.12: Modeling Variation
The power $P$ that can be obtained from a windmill is directly proportional to the cube of the wind speed $s$ . (a) Write an equation that expresses this variation. (Use $k$ for the constant of proportionality.) $\square$ (b) Find the constant of proportionality for a windmill that produces 128 watts of power when the wind is blowing at $20 \mathrm{mi} / \mathrm{h}$ . $k=$ $\square$ (c) Hoyv much power will this windmill produce if the wind speed increases to $30 \mathrm{mi} / \mathrm{h}$ ? $\square$ watts

Studdy Solution

STEP 1

What is this asking? We need to find a relationship between wind speed and power output for a windmill, figure out a constant based on given values, and then predict the power at a higher wind speed. Watch out! The problem says "cube," so we're dealing with the third power, not just multiplying by 3!
Also, make sure the units match up throughout the calculations.

STEP 2

1. Express the variation.
2. Find the constant.
3. Calculate the power at the new wind speed.

STEP 3

Alright, so we're told that power $P$ is directly proportional to the cube of wind speed $s$ .
This means $P$ increases as $s$ increases, but not at the same rate.
The "cube" tells us that we need to use $s^3$ .

STEP 4

Direct proportionality means we can write this relationship with a constant of proportionality, which we'll call $k$ .
So, our equation looks like this: $P = k \cdot s^3$

STEP 5

Now, we're given some real-world data!
We know that when the wind speed $s$ is 20 mi/h, the power $P$ is 128 watts.
Let's plug those values into our equation: $128 = k \cdot (20)^3$

STEP 6

Time to solve for $k$ !
First, let's calculate $20^3$ : $20^3 = 20 \cdot 20 \cdot 20 = 8000$ So our equation becomes: $128 = k \cdot 8000$

STEP 7

To isolate $k$ , we'll divide both sides of the equation by 8000: $\frac{128}{8000} = \frac{k \cdot 8000}{8000}$ $k = \frac{128}{8000} = 0.016$ So, our constant of proportionality, $k$ , is 0.016.

STEP 8

We now have our complete equation: $P = 0.016 \cdot s^3$ We want to find the power $P$ when the wind speed $s$ increases to 30 mi/h.
Let's plug that in: $P = 0.016 \cdot (30)^3$

STEP 9

First, we calculate $30^3$ : $30^3 = 30 \cdot 30 \cdot 30 = 27000$ Now, substitute this back into our equation: $P = 0.016 \cdot 27000$

STEP 10

Finally, multiply to find the power: $P = 432$ So, at 30 mi/h, the windmill will produce 432 watts of power!

STEP 11

(a) $P = k \cdot s^3$ (b) $k = 0.016$ (c) 432 watts

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Express the variation.
2. Find the constant.
3. Calculate the power at the new wind speed.

STEP 3

Alright, so we're told that power $P$ is directly proportional to the cube of wind speed $s$ .
This means $P$ increases as $s$ increases, but not at the same rate.
The "cube" tells us that we need to use $s^3$ .

STEP 4

Direct proportionality means we can write this relationship with a constant of proportionality, which we'll call $k$ .
So, our equation looks like this: $P = k \cdot s^3$

STEP 5

Now, we're given some real-world data!
We know that when the wind speed $s$ is 20 mi/h, the power $P$ is 128 watts.
Let's plug those values into our equation: $128 = k \cdot (20)^3$

STEP 6

Time to solve for $k$ !
First, let's calculate $20^3$ : $20^3 = 20 \cdot 20 \cdot 20 = 8000$ So our equation becomes: $128 = k \cdot 8000$

STEP 7

To isolate $k$ , we'll divide both sides of the equation by 8000: $\frac{128}{8000} = \frac{k \cdot 8000}{8000}$ $k = \frac{128}{8000} = 0.016$ So, our constant of proportionality, $k$ , is 0.016.

STEP 8

We now have our complete equation: $P = 0.016 \cdot s^3$ We want to find the power $P$ when the wind speed $s$ increases to 30 mi/h.
Let's plug that in: $P = 0.016 \cdot (30)^3$

STEP 9

First, we calculate $30^3$ : $30^3 = 30 \cdot 30 \cdot 30 = 27000$ Now, substitute this back into our equation: $P = 0.016 \cdot 27000$

STEP 10

Finally, multiply to find the power: $P = 432$ So, at 30 mi/h, the windmill will produce 432 watts of power!

STEP 11

(a) $P = k \cdot s^3$ (b) $k = 0.016$ (c) 432 watts

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Express the variation.2. Find the constant.3. Calculate the power at the new wind speed.

STEP 3

Alright, so we're told that power PPP is *directly proportional* to the *cube* of wind speed sss.This means PPP increases as sss increases, but not at the same rate.The "cube" tells us that we need to use s3s^3s3.

STEP 4

Direct proportionality means we can write this relationship with a constant of proportionality, which we'll call kkk.So, our equation looks like this: P=k⋅s3 P = k \cdot s^3 P=k⋅s3

STEP 5

Now, we're given some real-world data!We know that when the wind speed sss is **20 mi/h**, the power PPP is **128 watts**.Let's plug those values into our equation: 128=k⋅(20)3 128 = k \cdot (20)^3 128=k⋅(20)3

STEP 6

Time to solve for kkk!First, let's calculate 20320^3203: 203=20⋅20⋅20=8000 20^3 = 20 \cdot 20 \cdot 20 = 8000 203=20⋅20⋅20=8000 So our equation becomes: 128=k⋅8000 128 = k \cdot 8000 128=k⋅8000

STEP 7

To isolate kkk, we'll divide both sides of the equation by **8000**: 1288000=k⋅80008000 \frac{128}{8000} = \frac{k \cdot 8000}{8000} 8000128​=8000k⋅8000​ k=1288000=0.016 k = \frac{128}{8000} = 0.016 k=8000128​=0.016So, our constant of proportionality, kkk, is **0.016**.

STEP 8

We now have our complete equation: P=0.016⋅s3 P = 0.016 \cdot s^3 P=0.016⋅s3 We want to find the power PPP when the wind speed sss increases to **30 mi/h**.Let's plug that in: P=0.016⋅(30)3 P = 0.016 \cdot (30)^3 P=0.016⋅(30)3

STEP 9

First, we calculate 30330^3303: 303=30⋅30⋅30=27000 30^3 = 30 \cdot 30 \cdot 30 = 27000 303=30⋅30⋅30=27000 Now, substitute this back into our equation: P=0.016⋅27000 P = 0.016 \cdot 27000 P=0.016⋅27000

STEP 10

Finally, multiply to find the power: P=432 P = 432 P=432 So, at **30 mi/h**, the windmill will produce **432 watts** of power!

STEP 11

(a) P=k⋅s3 P = k \cdot s^3 P=k⋅s3 (b) k=0.016 k = 0.016 k=0.016 (c) 432 watts

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Express the variation.
2. Find the constant.
3. Calculate the power at the new wind speed.

Alright, so we're told that power $P$ is directly proportional to the cube of wind speed $s$ .
This means $P$ increases as $s$ increases, but not at the same rate.
The "cube" tells us that we need to use $s^3$ .

Direct proportionality means we can write this relationship with a constant of proportionality, which we'll call $k$ .
So, our equation looks like this: $P = k \cdot s^3$

Now, we're given some real-world data!
We know that when the wind speed $s$ is 20 mi/h, the power $P$ is 128 watts.
Let's plug those values into our equation: $128 = k \cdot (20)^3$

Time to solve for $k$ !
First, let's calculate $20^3$ : $20^3 = 20 \cdot 20 \cdot 20 = 8000$ So our equation becomes: $128 = k \cdot 8000$

To isolate $k$ , we'll divide both sides of the equation by 8000: $\frac{128}{8000} = \frac{k \cdot 8000}{8000}$ $k = \frac{128}{8000} = 0.016$ So, our constant of proportionality, $k$ , is 0.016.

We now have our complete equation: $P = 0.016 \cdot s^3$ We want to find the power $P$ when the wind speed $s$ increases to 30 mi/h.
Let's plug that in: $P = 0.016 \cdot (30)^3$

First, we calculate $30^3$ : $30^3 = 30 \cdot 30 \cdot 30 = 27000$ Now, substitute this back into our equation: $P = 0.016 \cdot 27000$

Finally, multiply to find the power: $P = 432$ So, at 30 mi/h, the windmill will produce 432 watts of power!

(a) $P = k \cdot s^3$ (b) $k = 0.016$ (c) 432 watts