*35 BRAINLY POINTS! BRAINLIEST TO THE FIRST CORRECT PERSON!*

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

3 years ago

Kendrick caught 22 rainbow trout and 18

liraira [26]

Answer:

16

Step-by-step explanation:

I think so I hope it helps

5 0

3 years ago

Which Property Is Shown By 2+5 = 5+2?

taurus [48]

I'm pretty sure it's commutative property of addition

6 0

3 years ago

Read 2 more answers

). If a, ß are zeroes of the quadratic polynomial p(x)=kx²+4x+4 such

natta225 [31]

Answer:

The values of k are 2/3 and -1

Step-by-step explanation:

Product of zeros = αβ= constant / coefficient of x^2 = 4/k

Sum of zeros =α+β = - coefficient of x / coefficient of x^2= -4/k

Given

Consider a= α and b= β

$(\alpha)^2 + (\beta)^2 = 24$

$(\alpha)^2 + (\beta)^2$ can be written as $(\alpha)^2 + 2(\alpha)(\beta) + (\beta)^2$ if we add $\pm 2 (\alpha)(\beta)$ in the above equation.

$(\alpha)^2 + 2(\alpha)(\beta) + (\beta)^2 -2(\alpha)(\beta)$

$(\alpha + \beta)^2 -2(\alpha)(\beta)$

Putting values of αβ and α+β

$(\frac {-4}{k})^2 -2( \frac {4}{k}) = 24\\\frac {16}{k^2} - \frac {8}{k} = 24\\Multiplying\,\, the \,\, equation\,\, with\,\, 8K^2\\ 2 - k= 3K^2\\3k^2-2+k=0\\or\\3k^2+k-2=0\\3k^2+3k-2k-2=0\\3k(k+1)-2(k+1)=0\\(3k-2)(k+1)=0\\3k-2=0 \,\,and\,\, k+1 =0\\k= 2/3 \,\,and\,\, k=-1$

The values of k are 2/3 and -1

8 0

3 years ago

IMPORTANT

EastWind [94]

$75 a month. 75x12 = 900. i used 12 because that’s the number of months in a year. hope this helps

7 0

3 years ago