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natka813 [3]
1 year ago
15

8 A clock in the shape of an hour glass is made up of two identical cones

Mathematics
1 answer:
ra1l [238]1 year ago
4 0

Answer:

The two cones are aligned on edge to edge so that the total volume of the hourglass becomes 2∗13πr2h

Step-by-step explanation:

a) When the sand has y units left in the upper cone the height of the cone becomes y, the radius remains 1cm.Now let's take the right triangle In the upper coneAnd by similar triangles property we found that the radius of the cone in which the amount of sand is left is y3 unitsNow the volume of the sand in the remaining part of the upper cone isπy32y which gives πy327Now if we apply the same similar triangle property with the cone in the below we find that the radius of the part it has filled up remains the same as y3unitsSo the volume it has filled is (3-y) from the top cone which says that the volume filled on the bottom cone has a height of (3-y)So therefore the volume of the frustum which it has filled up isπ3h(R2+r2+rR) {Is the volume of the frustum with two radii r and R with a height h}i.eπ3(3−y)(12+(y/3)2+(y/3))So now the total volume of the hourglass in terms of y is π3(3−y)(12+(y/3)2+(y/3))+πy327b) The next question tells us that when the height of the sand in the below cone reaches up a height of h centimetreswhich is exactly 3-y in the previous questionso substituting y = h-3 in the previous question we'll find the volume of the hourglass in terms of hπ(h−3)327+π3(6−h)(12+((h−3)/3)2+((h−3)/3)) .c) Total volume of the hourglass isV=V1+V2⇒3π4=πy327+π−π27(3−h)3Putting h=1We have the value of y as⇒3π4=π27y3+π+π278..............(1)⇒y=(54)13Differentiating the first equation with respect to t we have0=π9y2dydt+π9(3−h)2dhdtPutting values that dydt=−0.14inches/sec at h =1 and for ⇒y=(54)13We have0=π9(54)132∗(−0.14)+π922dhdt⇒0.1956π9=4π9dhdt⇒dhdt=0.19564⇒dhdt=0.0489

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Answer:

Objective Minimize 10x1 +15x2 + 12x3+18x4+15x5+ 21x6

The central total cost $ 17586 due to the number of central undergraduate students 1458 is very high.

The total minimum cost would $ 17586 +$260+ $300= $ 18146

Step-by-step explanation:

Let

X1 = # of undergraduate students from the East region,

X2 = # of graduate students from the East region,

X3 = # of undergraduate students from the Central region,

X4 = # of graduate students from the Central region,

X5 = # of undergraduate students from the West region, and

X6 = # of graduate students from the West region.

Then  the cost functions are

y1= 10x1 +15x2

y2= 12x3+18x4

y3= 15x5+ 21x6

According to the given conditions

The constraints are

x1 +x2 + x3+x4+ x5+ x6 ≥ 1500------- A

15X2 +18X4+21X6 ≥ 400---------B

21X6 ≥ 100

X6 ≥ 100/21

X6 ≥ 4.76

Taking

X6= 5

10X1 ≤ 500

X1 ≤ 500/10

X1≤ 5

18X4 ≥ 75

X4 ≥ 75/18

X4 ≥ 4.167

Taking

X4= 5

Putting the values

15X5+ 21X6 ≥ 300

15X5+ 21(5) ≥ 300

15X5+ 105 ≥ 300

15X5 ≥ 300-105

15X5 ≥ 195

X5 ≥ 195/15

X5 ≥ 13

Putting value of X6 and X4 in B

15X2 +18X4+21X6 ≥ 400

15X2 +18(5)+21(5) ≥ 400

15X2 +195 ≥ 400

15X2  ≥ 400-195

15X2  ≥ 205

X2  ≥ 205/15

X2  ≥ 13.67

Taking X2= 14

Now putting the values in the cost equations to check whether the conditions are satisfied.

y1= 10x1 +15x2

y1= 10 (5) + 15(14)= 50 + 210= $ 260

y3= 15x5+ 21x6

y3= 15 (13) + 21(5)

y3= 195+105= $ 300

x1 +x2 + x3+x4+ x5+ x6 ≥ 1500

5+14+x3+5+13+5≥ 1500

x3≥ 1500-42

x3≥ 1458

y2= 12x3+18x4

y2= 12 (1458)  + 18 (5)

y2= 17496 +90

y2= $ 17586

The cost can be minimized if the number of students from

                                 Undergraduate         Graduate

East Region              X1≤ 5                            X2  ≥ 13.67

Central                      X3≥ 1458                   X4 ≥ 4.167

West                          X5 ≥ 13                          X6 ≥ 4.76

This will result in the required number of students that is 1500

Constraints:

East Undergraduate must not be greater than 5

East Graduate must not be less than 13

Central Undergraduate must  be greater than 1458

Central Graduate must  be greater than 4

West Undergraduate must  be greater than 13

West Graduate must  be greater than 4

The central total cost $ 17586 due to the number of central undergraduate students 1458 is very high.

The east region has a least cost of $260 and west region has a cost of $300.

The total minimum cost would $ 17586 +$260+ $300= $ 18146

6 0
3 years ago
Maggie’s brother is three years younger than twice her age. The sum of their ages is 24. How old is maggie?
JulsSmile [24]

To solve this problem, we must set up a system of equations. In this case, let's let Maggie's age be represented by the variable m and her brother's age be represented by the variable b. We are told that the sum of their ages is 24, which gives us our first equation: m + b = 24. We can construct our next equation from the first sentence of given information: b = 2m - 3. This makes our system of equations:

m + b = 24

b = 2m - 3

To solve, we are going to substitute the value for b in terms of m given by the second equation into the first equation for the variable b.

m + b = 24

m + 2m - 3 = 24

To simplify, we must first combine the variable terms on the left side of the equation using addition.

3m - 3 = 24

Next, we should add 3 to both sides of the equation to get the variable term alone on the left side of the equation.

3m = 27

Finally, we should divide both sides by 3 in order to get the variable m alone.

m = 9

Therefore, Maggie is 9 years old (using the first equation and substituting in this value you can find that her brother is 15 years old).

Hope this helps!

8 0
3 years ago
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A person earns $43200.00 one year and receives an 8% raise in salary. what is the new salary?
inysia [295]
46,656. 43,200 x 1.08
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3 years ago
Write each of the following decimals in words.
Sergeeva-Olga [200]

A. nine hundredths

B. Thirty five hundredths

C. Seven Tenths

D. Three Thousandths

E. One Hundred Fourty five Thousandths

F. Fifty Nine Ten Thousandths

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Breana is selling tickets to a choral performance. On the first day of ticket sales , she sold 3 senior citizen tickets and 1 ch
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Answer:

The senior tickets are 9 dollars and 30 cents.

Step-by-step explanation:


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