Maths

A rectangular box of internal base dimensions 2r, 3r and internal height 9r is filled with spheres of radius r. Find the maximum number of spheres that can be fitted into the box.

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Contributor-Level 9

Let L be the vertical distance between the centers of two adjacent balls.

$L = \sqrt[]{[{(2 r)}^{2} - r^{2}]} = r \sqrt[]{3}$

Also, if the box can hold n sphere then (n – 1) L + 2r

≤ 9r [ the last ball height = 2r]

$(n - 1) \sqrt[]{3} r \leq 7 r$

$n \leq \frac{7}{\sqrt[]{3}} + 1$

n ≤ 5.04

Since, n is a natural number.

n = 5

So, the box can hold maximum of 5 spheres.

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7 months ago

An pipe can fill a tank in 3 hours. Because of a leak in the tank it was taking 3.5 hrs to fill the tank. Find the time in which the leak can drain all the water of the tank when full.

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Contributor-Level 10

(b):Total time required to empty it =

$\frac{1}{(\frac{1}{3} - \frac{1}{3.5})} = 21 hours.$

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7 months ago

Answer the questions based on the following information.

Mr. John has 5 girls and 9 boys. He decides to give them a present on the occasion of Christmas. He takes 200 one-rupee coins and puts them in 2 bags which he closes. One of the bags contains more coins than the other. He asks his boys to choose one of the bags. They chose the bag with more money. When money is divided equally among boys they are left with Rs. 3. When money is divided equally among girls (not necessarily the same amount as given to boys), Rs. 4 was left. No girl gets less than a boy.

The total number of coins in the bag with fewer coins is

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Contributor-Level 9

9B + 3 + 5G + 4 = 200 9B + 5G = 193

The positive integer solutions are (B, G) = (2, 35) (7, 25) (12, 17) (17, 8)

Only one of the (B, G) gives G > B and 9B + 3 > 100, i.e. (B, G) = (12, 17)

17 * 5 = 85

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7 months ago

I can row at the rate of 8 miles per hour in still water. It takes twice as long to row up a river as to row down the river. Find the speed at which the river flows.

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Contributor-Level 10

(a):Let, s be the speed of stream.

So, 8 + s = 2 (8 – s)

s = 8/3 mph

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7 months ago

Find the coordinates of the centroid G of a triangle with vertices at (3, 7), (5, 5) and (–3, 2).

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Contributor-Level 10

(a):Let G be (X, Y) then $X = \frac{{3 + 5 + (- 3)}}{3} = \frac{5}{3}$ $\frac{}{} \frac{}{}$

$Y = \frac{(7 + 5 + 2)}{3} = \frac{14}{3}$

$\Rightarrow G is (\frac{5}{3}, \frac{14}{3})$

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7 months ago

Answer the questions based on the following information.

The difference between the final share of a girl and a boy was

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Contributor-Level 9

9B + 3 + 5G + 4 = 200 9B + 5G = 193

The positive integer solutions are (B, G) = (2, 35) (7, 25) (12, 17) (17, 8)

Only one of the (B, G) gives G > B and 9B + 3 > 100, i.e. (B, G) = (12, 17)

17 – 12 = 5

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7 months ago

The average score of 20 candidates in an exam is 25 marks. It decreases by 2 marks when the score of the topper is excluded. What are the marks obtained by the topper?

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Contributor-Level 10

Average score of 20 candidate = 25 marks

Total score of 20 candidate = 500 marks

Let the score of the topper be x. Then,

$\frac{(500 - x)}{19} = 23$

500 – x = 437

x = 500 – 437 = 63 marks

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New answer posted

7 months ago

Answer the questions based on the following information.

How many coins more did the bag, which the boys got, contain than the oth

...more

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Contributor-Level 9

9B + 3 + 5G + 4 = 200 9B + 5G = 193

The positive integer solutions are (B, G) = (2, 35) (7, 25) (12, 17) (17, 8)

Only one of the (B, G) gives G > B and 9B + 3 > 100, i.e. (B, G) = (12, 17)

12 * 9 – 17 * 5 = 23

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7 months ago

Let f defined as f (3n) = n + f (3n – 3), where n is a positive integer greater than 1. If f (3n) = 1 when n=1, then find last digit of f (3¹²)

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Payal Gupta

Contributor-Level 10

f (3n) – f (3n– 3) = n

n = 1

f (3) – f (0) = 1

f (0) = 0

n = 2

$\begin{array}{l} f (6) - f (3) = 2 \\ ? ? ? ? \end{array}$

f (3n) – f (3) = 2 + 3 + 4+…………….+ n

f (3n) = 1 + 2 + 3 +…………….+ n

= $n \frac{(n + 1)}{2}$

$?$ f (1312) – f (3 * 311)

$\frac{3^{11}}{2} (3^{11} + 1) = \frac{3^{22} + 3^{11}}{2}$

Hence, remainder $[\frac{\frac{3^{22} + 3^{11}}{2}}{10}]$ = 8

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7 months ago