What is the missing step in this proof

The surface area of a cylinder is increasing at a rate of 9 pi square meters per hour. The height of the cylinder is fixed at 3

Alekssandra [29.7K]

Answer:

$9\pi \text{ cubic meters per hour}$

Step-by-step explanation:

Since, the surface area of a cylinder,

$A= 2\pi r^2 + 2\pi rh$ ................(1)

Where,

r = radius,

h = height,

If $A= 36\pi\text{ square meters}, h = 3\text{ meters}$

$36\pi = 2\pi r^2 + 2\pi r(3)$

$18 = r^2 + 3r$

$\implies r^2 + 3r - 18=0$

$r^2 + 6r - 3r - 18 = 0$ ( by middle term splitting )

$r(r+6)-3(r+6)=0$

$(r-3)(r+6)=0$

By zero product property,

r = 3 or r = - 6 ( not possible )

Thus, radius, r = 3 meters,

Now, differentiating equation (1) with respect to t ( time ),

$\frac{dA}{dt}= 4\pi r\frac{dr}{dt} +2\pi(r\frac{dh}{dt} + h\frac{dr}{dt})$

∵ h = constant, ⇒ dh/dt = 0,

$\frac{dA}{dt} = 4\pi r \frac{dr}{dt} +2\pi h \frac{dr}{dt}$

We have, $\frac{dA}{dt}=9\pi\text{ square meters per hour}, r = h = 3\text{ meters}$

$9\pi = 4\pi (3) \frac{dr}{dt}+2\pi (3)\frac{dr}{dt}$

$9\pi = (12\pi + 6\pi )\frac{dr}{dt}$

$9\pi = 18\pi \frac{dr}{dt}$

$\implies \frac{dr}{dt} =\frac{1}{2}\text{ meter per hour}$

Now,

Volume of a cylinder,

$V=\pi r^2 h$

Differentiating w. r. t. t,

$\frac{dV}{dt}=\pi ( r^2 \frac{dh}{dt}+h(2r)\frac{dr}{dt})=\pi ((3)(6) (\frac{1}{2})) = 9\pi \text{ cubic meters per hour}$

6 0

3 years ago

The difference of 48 and a number

scoundrel [369]

48-X or C-48 would be the answer

3 0

3 years ago

Sam have worked these hours during the week: 4.5, 8.75, 9.5, 10, and 4.25 hours. How many hours did Sam work?

ziro4ka [17]

Answer:

37 hours

Step-by-step explanation:

4.5 + 8.75 + 9.5 + 10 + 4.25 = 37 hours

3 0

3 years ago

Read 2 more answers

I need help with this question. Can you please help me. I’ll give you 18 points if it’s correct

lukranit [14]

Answer:

34.4

Step-by-step explanation:

Using Triangle Sum Theory, you see that the triangles are similar. They have the same angle measurements. That means their corresponding sides are proportional.

$\frac{MN}{NO}$ = $\frac{PQ}{QR}$

$\frac{14}{11}$ = $\frac{PQ}{27}$

Cross multiply

14(27) = 11(PQ)

378 = 11(PQ)

$\frac{378}{11}$ = PQ

PQ = 34.4

7 0

3 years ago

Adult tickets to a play cost $22 tickets for kids cost $15. Tickets for a group of 11 cost a total of $228. How many children an

Zina [86]

Answer:Number of children = 2 and number of adults = 9 attended the play

Step-by-step explanation:

Step 1

Let the number of children be represented as x

and the number of adults in the group be represented as as y

such that the total number of the group that went to the play

= x+y=11--- equation 1

And the total cost paid for the play will be expressed as

15x+ 22y= 228------ equation 2

Step 2-- Solving

x+y=11--- equation 1

15x+ 22y= 228------ equation 2

B y elimination method, multiply equation 1 by 15 to give equation 3 and then subtract from equation 2

15x+ 22y= 228------ equation 2

15x+15Y= 165----- equation 3

7y=63

y= 63/7= 9

y= adults = 9

therefore x = number of children will be

x+y=11

x+ 9=11

x= 11-9= 2

In the group that attended the play, Number of children = 2 and number of adults = 9

8 0

3 years ago

What is the missing step in this proof￼

What is the missing step in this proof