Math / Calculus

QuestionA rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?
Change in circumference $=$

Studdy Solution

STEP 1

What is this asking? If a ripple grows outwards at a speed of $4$ feet per second, how quickly is the outer edge of the ripple growing when the ripple has a radius of $15$ feet? Watch out! We're dealing with rates of change here, so we'll need some calculus!
Don't mix up radius and circumference.

STEP 2

1. Define the relationship between circumference and radius.
2. Differentiate with respect to time.
3. Plug in the given values.
4. Calculate the result.

STEP 3

We know that the circumference $C$ of a circle is related to its radius $r$ by the formula $C = 2 \cdot \pi \cdot r$ .
This is our starting point!

STEP 4

We want to find how fast the circumference is changing with respect to time, so we'll differentiate both sides of our equation with respect to $t$ (time).
Remember, both $C$ and $r$ are functions of time!

STEP 5

Differentiating $C = 2 \cdot \pi \cdot r$ with respect to $t$ gives us $\frac{dC}{dt} = 2 \cdot \pi \cdot \frac{dr}{dt}$ .
Remember, the derivative of $r$ with respect to $t$ is $\frac{dr}{dt}$ , and $2 \cdot \pi$ is a constant that comes along for the ride!

STEP 6

We're given that the radius is increasing at a rate of $4$ feet per second.
This means $\frac{dr}{dt} = 4$ .
We want to find $\frac{dC}{dt}$ when $r = 15$ , but notice that $r$ doesn't even appear in our differentiated equation!
That makes things easier!

STEP 7

Substituting $\frac{dr}{dt} = 4$ into our equation, we get $\frac{dC}{dt} = 2 \cdot \pi \cdot 4$ .

STEP 8

Now, we just multiply! $\frac{dC}{dt} = 8\pi$ .
So, the circumference is changing at a rate of $8\pi$ feet per second.
Awesome!

STEP 9

When the radius is $15$ feet, the circumference is changing at a rate of $8\pi$ feet per second.

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QuestionA rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?
Change in circumference $=$

Studdy Solution

STEP 1

What is this asking? If a ripple grows outwards at a speed of $4$ feet per second, how quickly is the outer edge of the ripple growing when the ripple has a radius of $15$ feet? Watch out! We're dealing with rates of change here, so we'll need some calculus!
Don't mix up radius and circumference.

STEP 2

1. Define the relationship between circumference and radius.
2. Differentiate with respect to time.
3. Plug in the given values.
4. Calculate the result.

STEP 3

We know that the circumference $C$ of a circle is related to its radius $r$ by the formula $C = 2 \cdot \pi \cdot r$ .
This is our starting point!

STEP 4

We want to find how fast the circumference is changing with respect to time, so we'll differentiate both sides of our equation with respect to $t$ (time).
Remember, both $C$ and $r$ are functions of time!

STEP 5

Differentiating $C = 2 \cdot \pi \cdot r$ with respect to $t$ gives us $\frac{dC}{dt} = 2 \cdot \pi \cdot \frac{dr}{dt}$ .
Remember, the derivative of $r$ with respect to $t$ is $\frac{dr}{dt}$ , and $2 \cdot \pi$ is a constant that comes along for the ride!

STEP 6

We're given that the radius is increasing at a rate of $4$ feet per second.
This means $\frac{dr}{dt} = 4$ .
We want to find $\frac{dC}{dt}$ when $r = 15$ , but notice that $r$ doesn't even appear in our differentiated equation!
That makes things easier!

STEP 7

Substituting $\frac{dr}{dt} = 4$ into our equation, we get $\frac{dC}{dt} = 2 \cdot \pi \cdot 4$ .

STEP 8

Now, we just multiply! $\frac{dC}{dt} = 8\pi$ .
So, the circumference is changing at a rate of $8\pi$ feet per second.
Awesome!

STEP 9

When the radius is $15$ feet, the circumference is changing at a rate of $8\pi$ feet per second.

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QuestionA rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?Change in circumference ===

Studdy Solution

STEP 1

STEP 2

1. Define the relationship between circumference and radius.2. Differentiate with respect to time.3. Plug in the given values.4. Calculate the result.

STEP 3

We know that the circumference CCC of a circle is related to its radius rrr by the formula C=2⋅π⋅rC = 2 \cdot \pi \cdot rC=2⋅π⋅r.This is our **starting point**!

STEP 4

We want to find how fast the circumference is changing *with respect to time*, so we'll differentiate both sides of our equation with respect to ttt (time).Remember, both CCC and rrr are functions of time!

STEP 5

STEP 6

We're given that the radius is increasing at a rate of **444 feet per second**.This means drdt=4\frac{dr}{dt} = 4dtdr​=4.We want to find dCdt\frac{dC}{dt}dtdC​ when r=15r = 15r=15, but notice that rrr doesn't even appear in our differentiated equation!That makes things easier!

STEP 7

Substituting drdt=4\frac{dr}{dt} = 4dtdr​=4 into our equation, we get dCdt=2⋅π⋅4\frac{dC}{dt} = 2 \cdot \pi \cdot 4dtdC​=2⋅π⋅4.

STEP 8

Now, we just multiply! dCdt=8π\frac{dC}{dt} = 8\pidtdC​=8π.So, the circumference is changing at a rate of 8π8\pi8π feet per second.Awesome!

STEP 9

When the radius is 151515 feet, the circumference is changing at a rate of 8π8\pi8π feet per second.

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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.

QuestionA rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?
Change in circumference $=$

1. Define the relationship between circumference and radius.
2. Differentiate with respect to time.
3. Plug in the given values.
4. Calculate the result.

We know that the circumference $C$ of a circle is related to its radius $r$ by the formula $C = 2 \cdot \pi \cdot r$ .
This is our starting point!

We want to find how fast the circumference is changing with respect to time, so we'll differentiate both sides of our equation with respect to $t$ (time).
Remember, both $C$ and $r$ are functions of time!

We're given that the radius is increasing at a rate of $4$ feet per second.
This means $\frac{dr}{dt} = 4$ .
We want to find $\frac{dC}{dt}$ when $r = 15$ , but notice that $r$ doesn't even appear in our differentiated equation!
That makes things easier!

Substituting $\frac{dr}{dt} = 4$ into our equation, we get $\frac{dC}{dt} = 2 \cdot \pi \cdot 4$ .

Now, we just multiply! $\frac{dC}{dt} = 8\pi$ .
So, the circumference is changing at a rate of $8\pi$ feet per second.
Awesome!

When the radius is $15$ feet, the circumference is changing at a rate of $8\pi$ feet per second.