A. y = -1.33x; 2.67<br> B. y = -0.75x; 1.50<br> C. y = 1.33x; -2.67<br> D. y = 0.13x; -0.25

A punch glass is in the shape of a hemisphere with a radius of 5 cm. If the punch is being poured into the glass so that the cha

Galina-37 [17]

Answer:

28.27 cm/s

Step-by-step explanation:

Though Process:

The punch glass (call it bowl to have a shape in mind) is in the shape of a hemisphere
the radius $r=5cm$
Punch is being poured into the bowl
The height at which the punch is increasing in the bowl is $\frac{dh}{dt} = 1.5$
the exposed area is a circle, (since the bowl is a hemisphere)
the radius of this circle can be written as $'a'$

what is being asked is the rate of change of the exposed area when the height $h = 2 cm$
the rate of change of exposed area can be written as $\frac{dA}{dt}$ .
since the exposed area is changing with respect to the height of punch. We can use the chain rule: $\frac{dA}{dt} = \frac{dA}{dh} . \frac{dh}{dt}$

and since $A = \pi a^2$ the chain rule above can simplified to $\frac{da}{dt} = \frac{da}{dh} . \frac{dh}{dt}$ -- we can call this Eq(1)

Solution:

the area of the exposed circle is

$A =\pi a^2$

the rate of change of this area can be, (using chain rule)

$\frac{dA}{dt} = 2 \pi a \frac{da}{dt}$ we can call this Eq(2)

what we are really concerned about is how $a$ changes as the punch is being poured into the bowl i.e $\frac{da}{dh}$

So we need another formula: Using the property of hemispheres and pythagoras theorem, we can use:

$r = \frac{a^2 + h^2}{2h}$

and rearrage the formula so that a is the subject:

$a^2 = 2rh - h^2$

now we can derivate a with respect to h to get $\frac{da}{dh}$

$2a \frac{da}{dh} = 2r - 2h$

simplify

$\frac{da}{dh} = \frac{r-h}{a}$

we can put this in Eq(1) in place of $\frac{da}{dh}$

$\frac{da}{dt} = \frac{r-h}{a} . \frac{dh}{dt}$

and since we know $\frac{dh}{dt} = 1.5$

$\frac{da}{dt} = \frac{(r-h)(1.5)}{a}$

and now we use substitute this $\frac{da}{dt}$ . in Eq(2)

$\frac{dA}{dt} = 2 \pi a \frac{(r-h)(1.5)}{a}$

simplify,

$\frac{dA}{dt} = 3 \pi (r-h)$

This is the rate of change of area, this is being asked in the quesiton!

Finally, we can put our known values:

$r = 5cm$

$h = 2cm$ from the question

$\frac{dA}{dt} = 3 \pi (5-2)$

$\frac{dA}{dt} = 9 \pi cm/s// or//\frac{dA}{dt} = 28.27 cm/s$

5 0

2 years ago

How would 350 percent be written as a simplified fraction

Sauron [17]

Divide by 100. 350/100=3.5 which is 3 and 1/2

4 0

3 years ago

Read 2 more answers

Help help help help help

dem82 [27]

I believe it should be B because it says one more than twice the number so that means addition. Also it says twice the number so that should be 2w.
I’m not 100% sure but it is my best guess

7 0

3 years ago

14 ) Mark Achin sells 3,600 electric motors each year . The cost of these is $ 200 each , and demand is constant throughout the

ANTONII [103]

Mark Achin's economic order quantity in units is <u>D. 120 units</u>.

<h3>What is economic order quantity?</h3>

The economic order quantity (EOQ) is the optimal order quantity that minimizes the total costs (ordering, receiving, and holding inventory).

The formula for the economic order quantity is the square root of [2(setup costs)(demand rate)] / holding costs.

<h3>Data and Calculations:</h3>

Annual demand = 3,600

Cost price per unit = $200

Ordering cost = $40

Holding cost per unit = $20

Working days per year = 360

Lead time = 5 days

Economic order quantity (EOQ) = square root of: [2(setup costs)(demand rate)] / holding costs.

= square root of (2 x $40 x 3,600)/$20

= 120

Thus, Mark Achin's economic order quantity in units is <u>D. 120 units</u>.

Learn more about the economic order quantity at brainly.com/question/13386271

#SPJ1

3 0

2 years ago

Can you show your work 12/2=9/?

timama [110]

Since this is a proportion, we cross multiply.
12/2=9/x (The variable "x" is used the represent the unknown number, and this is the number we are solving for)
12*x=2*9 (Cross multiply)
12x=18 (Simplify)
x=18/12 (Divide each side by 12)
x = 3/2 (Simplify fraction)
The unknown number is 3/2, or 1.5.

3 0

2 years ago

Read 2 more answers

A. y = -1.33x; 2.67 B. y = -0.75x; 1.50 C. y = 1.33x; -2.67 D. y = 0.13x; -0.25