Question

Each side of a square is increasing at a rate of 6cm/s. At what rate is the area of the square increasing when the area of the square is 16cm^(2)

Please show the steps so I can see how to answer questions such as these

Answer 1

The answer is 48cm^(2) I'm not even getting results close to this

Answer 2

Let x = length of side of the square. Then area of the square is
A(x) = x^2

You are also told that dx/dt = 6 cm/s

Now, you want to find dA/dt when A = 16, i.e., when x = 4.

By the chain rule
\[ \frac{dA}{dt} = \frac{dA}{dx}\frac{dx}{dt} \]

Try that.

Answer 3

I'm still lost I understand what dA/dt is but I just dont get how to solve it
dA/dt A' = 2xx'
16/2(6) = x'
x' = 16/12

which is incorrect what am I doing wrong.

Answer 4

yeah I'm still completely lost

Answer 5

where are you getting x = 4 from

Answer 6

Ok I get it now

Answer 7

so tell me

Answer 8

Since
dx/dt = 6cm/s
and
dA/dx = 16cm/s

dA/dx dx/dt
= dA/dt

dA/dt A' = 2xx'
A' = 2x6cm/s

So we need to solve for x because we are trying to find dA
so
16 = x^(2)
(16)^(1/2) = 4

thus

A' = 2(4)(6)
A' = 48

Answer 9

Thanks for the help, I think I finally understand d/dx now

Answer 10

No, no. Wait.

Answer 11

don't worry I'm not going anywhere did I do something wrong?

Answer 12

By the chain rule
\[ \frac{dA}{dt} = \frac{dA}{dx} \frac{dx}{dt} \]

Look at the function for A, it is A(x) = x^2. Hence dA/dx = 2x.

Further, you are told that dx/dt = 6. Therefore using the equation above
\[ \frac{dA}{dt} = \frac{dA}{dx} \frac{dx}{dt} = 2x . 6 = 12x \]

Answer 13

so far so good?

Answer 14

yes

Answer 15

When the area of the square is 16, then x = 4. Hence instantaneously, at the moment,
\[ \frac{dA}{dt} = 12x = 12(4) = 48 \]