Math / Algebra

Question(1 point)
The area of a circular wave expands across a still pond such that its radius increases by 4 cm each second. Write a formula for the area $A$ of the circle as a function of time $t$ since the wave begins: $A=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find a formula that tells us the area of a circular wave at any given time, knowing that the wave's radius grows by 4 cm every second. Watch out! Don't forget that the problem asks for the area as a function of time, not radius.
We'll need to connect the radius to time somehow.

STEP 2

1. Relate radius and time
2. Express area in terms of time

STEP 3

Alright, so the radius increases by $4$ cm every second.
This means that after $t$ seconds, the radius will have increased by $4 \cdot t$ cm.
Since the wave starts as a tiny point, we can say the initial radius is $0$ cm.
Therefore, the radius $r$ at any time $t$ is given by $r = 4 \cdot t$ cm.
Boom!

STEP 4

We know that the area $A$ of a circle with radius $r$ is given by the formula $A = \pi \cdot r^2$ .
This is our key to victory!

STEP 5

Now, let's substitute our expression for the radius in terms of time, $r = 4 \cdot t$ , into the area formula.
This gives us $A = \pi \cdot (4 \cdot t)^2$ .
We're so close, we can taste it!

STEP 6

Let's simplify this expression.
We have $A = \pi \cdot (4 \cdot t)^2 = \pi \cdot (16 \cdot t^2) = 16 \cdot \pi \cdot t^2$ .
Fantastic!

STEP 7

The formula for the area $A$ of the circle as a function of time $t$ is $A = 16 \cdot \pi \cdot t^2$ .

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Question(1 point)
The area of a circular wave expands across a still pond such that its radius increases by 4 cm each second. Write a formula for the area $A$ of the circle as a function of time $t$ since the wave begins: $A=$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Relate radius and time
2. Express area in terms of time

STEP 3

STEP 4

We know that the area $A$ of a circle with radius $r$ is given by the formula $A = \pi \cdot r^2$ .
This is our key to victory!

STEP 5

Now, let's substitute our expression for the radius in terms of time, $r = 4 \cdot t$ , into the area formula.
This gives us $A = \pi \cdot (4 \cdot t)^2$ .
We're so close, we can taste it!

STEP 6

Let's simplify this expression.
We have $A = \pi \cdot (4 \cdot t)^2 = \pi \cdot (16 \cdot t^2) = 16 \cdot \pi \cdot t^2$ .
Fantastic!

STEP 7

The formula for the area $A$ of the circle as a function of time $t$ is $A = 16 \cdot \pi \cdot t^2$ .

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Question(1 point)The area of a circular wave expands across a still pond such that its radius increases by 4 cm each second. Write a formula for the area AAA of the circle as a function of time ttt since the wave begins: A=A=A= □\square□

Studdy Solution

STEP 1

STEP 2

1. Relate radius and time2. Express area in terms of time

STEP 3

STEP 4

We know that the area AAA of a circle with radius rrr is given by the formula A=π⋅r2A = \pi \cdot r^2A=π⋅r2.This is our key to victory!

STEP 5

Now, let's **substitute** our expression for the radius in terms of time, r=4⋅tr = 4 \cdot tr=4⋅t, into the area formula.This gives us A=π⋅(4⋅t)2A = \pi \cdot (4 \cdot t)^2A=π⋅(4⋅t)2.We're so close, we can taste it!

STEP 6

Let's **simplify** this expression.We have A=π⋅(4⋅t)2=π⋅(16⋅t2)=16⋅π⋅t2A = \pi \cdot (4 \cdot t)^2 = \pi \cdot (16 \cdot t^2) = 16 \cdot \pi \cdot t^2A=π⋅(4⋅t)2=π⋅(16⋅t2)=16⋅π⋅t2.Fantastic!

STEP 7

The formula for the area AAA of the circle as a function of time ttt is A=16⋅π⋅t2A = 16 \cdot \pi \cdot t^2A=16⋅π⋅t2.

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Question(1 point)
The area of a circular wave expands across a still pond such that its radius increases by 4 cm each second. Write a formula for the area $A$ of the circle as a function of time $t$ since the wave begins: $A=$ $\square$

1. Relate radius and time
2. Express area in terms of time

We know that the area $A$ of a circle with radius $r$ is given by the formula $A = \pi \cdot r^2$ .
This is our key to victory!

Now, let's substitute our expression for the radius in terms of time, $r = 4 \cdot t$ , into the area formula.
This gives us $A = \pi \cdot (4 \cdot t)^2$ .
We're so close, we can taste it!

Let's simplify this expression.
We have $A = \pi \cdot (4 \cdot t)^2 = \pi \cdot (16 \cdot t^2) = 16 \cdot \pi \cdot t^2$ .
Fantastic!

The formula for the area $A$ of the circle as a function of time $t$ is $A = 16 \cdot \pi \cdot t^2$ .