How the pressure depending on depth underwater

 A reader send me this problem:

When diving in the ocean, you must consider how much pressure you will experience from diving a certain depth. From the atmosphere, we experience 14.7 pounds per square inch (psi) and for every foot we dive down into the ocean, we experience another 0.44 psi in pressure.

a.      Write a function expressing how pressure changes depending on depth underwater.

b.     How far can you dive without experiencing more than 58.7 psi of pressure on your body?

 

 

My dear, regarding your problem look at my explanation:

 

a)     For each foot in depth the pressure increases by 0.44 psi / foot:

At the surface p = p 0

1 foot in depth  p 1 = p 0 + 0.44 (psi)

2 feet in depth p 2 = p 0 + 2 * 0.44 (psi)

3 feet in depth p 3 = p 0 + 3 *0,44 (psi)

….

X feet in depth p = p 0 + X * 0.44 (psi)

The function is p (X) = 14.7 + X * 0.44

 

b)     When the pressure in 58.5 psi, what is the depth?

P (X) = 58.5 = 14.7 + X * 0.44

58.5 – 14.7 = X * 0.44

43.8 = X * 0.44

X = 43.8 : 0.44

X = 99.54 feet

Test 2

1.

Aamir wants to print some brochures for his business. If he goes for one style, it will cost him Rs. 125 plus Rs. 0.2 per brochure. If he goes for the other style, it will cost him Rs. 25 plus Rs. 0.45 per brochure. For how many brochures will the price for both styles be the same?

Solution:

You can understand from the text of the problem that Aamir is printing a number “x” of brochures for which the price will be the same for both styles. Next, you need to note the price with “P”. The price “P” paid for printing the brochures in the first style is:
P = 125 + 0.2 * x

The price “P” paid for printing the brochures in the second style is:

P = 25 + 0.45 *

As you equal the equations, you obtain:

125 + 0.2 * x = 25 + 0.45 * x

125 – 25 = 0.45 * x – 0.2 * x

100 = 0.25 * x

x = 100 / 0.25 = 400

2.

Pointing to the girl playing cricket, Matthew says, “Her mother is the only daughter of my mother-in-law”. Who is Matthew to the girl?

Solution:

The mother-in-law is the mother of his wife, so the only daughter of his mother-in-law is his wife. The girl is the daughter of his wife, so Matthew is the father of the girl.

3.

Find the missing number:

6	10	10	26
8	4	14	26
8	4	14	26
6	3	17	26
7	5	14	26

Solution:

We observe that the sum of the numbers from each row is 26, so, on the second row, the sum needs to be 26.

8 + 4 + x = 26 , so x = 26 – 8 – 4 = 14

4.

Sita and Ammir participated in college election. Sita got 61 % votes and won by a margin of 176 votes. Find total number of votes.

Solution:

Sita got 61% of the votes, so Aamir got 39% of the votes. The difference between their procents is 61 – 39 = 22%, which represent 176 votes. The total number of votes, 100%, is:

22% . . . . . 176

100% . . . . . x

x = 100 * 176 / 22 = 800 (votes)

5.

Find the area of un-shaded part (area of each small square is 1 cm²).

Solution:

The shaded surface has 5 small squares and 4 triangles, each triangle representing half of a small square. The whole shaded surface has the area of 5 + 4 * 0.5 = 7 cm².
The area of the whole rectangle is 5 * 3 = 15 cm².

The unshaded area is 15 – 7 = 8 cm².

6.

Balvinder can finish a work in 28 minutes. Saina works thrice as fast as Balvinder. How long will it take to finish the work, if Balvinder and Saina work together?

Solution:

You’ll introduce the notion of work speed, which represents the speed with which each of them is working, and you’ll note it with S(B), which is Balvinder’s work speed, and S(S), which is Saina’s work speed. Also, you’ll obtain S(S) = 3 * S(B).

The work is finished by Balvider in 28 minutes.

work = 28 * S(B)

When Balvinder and Saina are working together, the two speeds will sum, and the work is finished in time “t”.

work = t * [S(B) + S(S)]

work = t * [S(B) + 3 * S(B)]

work = t * 4 * S(B), but also work = 28 * S(B)

Now you’ll equal them, and you’ll obtain:

28 * S(B) = t * 4 * S(B)

T = [28 * S(B)] / [4 * S(B)] = 7 , so the work is finished in 7 minutes.

7.

The number of hours left in the day are half of hours already passed. How many hours have already passed in the day?

Solution:

A day has 24 hours. You note the number of hours left with “x”, so the number of hours passed will become 2 * x. Now you obtain the equation:

2 * x + x = 24

3 * x = 24

X = 24 / 3 = 8 , so there are 8 hours left of the day.

You’ll now obtain 24 – 8 = 16 hours that already passed.

Test

It is a good time for new post because a young student from Pakistan sent a test asking me the solution.

You could still see the question and the solution. 

1. 

Mr. and Mrs. Singh went to the magic show with their 3 children. The children were charged half the entrance fee. If the total fee charged was Rs.42, what is the entrance fee for each adult?

Solution:

They paid Rs 42 for 2 full entrance fees (tickets) and 3 halfs of an entrance fee. That means 3 and a half tickets, so the price of a full ticket (adult entrance fee) is Rs 42/3,5, which means Rs 12.

2.

 If we use (x+) to indicate the following sum

1+2+3+....+x

then find the value of k in the following equation

(21+) - (20+) = (k+)

Solution:

1    If you use (x+) to indicate the sum of the natural numbers from 1 to x, then you can rewrite the equation. Instead of (21+) – (20+), you’ll have (1+2+…+20+21) – (1+2+…+20) = 1+2+…+20+21-1-2-…-20 = 21. Now you need to find the k which resolves the equation: (k+) = 21. Here, you can apply Gauss’s sum, which says that (1+2+…+k) = [k*(k+1)]/2. Now [k*(k+1)]/2 = 21, and you multiply by 2, so k*(k+1) = 42. Which are the consecutive natural numbers that multiplied equal 42? k=6 and k+1=7.

3. 

Find total length of lines shown in the picture (all dimensions are in meters).

Solution:
The figure is a rectangle, which has the property that the opposite sides are parallel and equal, two by two, so the other 2 sides of the rectangle have the lengths of 108 and 144 meters. The fifth line is one of the diagonals of the rectangle, whose length can be calculated with Pythagorean theorem. This theorem says that the square of the hypotenuse equals to the sum of the squares of the two others sides, so 108^2 + 144^2 = x^2. You calculate the sum and you obtain 32400, so x^2 = 32400. x is the square root of 32400, which means 180. Now you can calculate the sum of all the lines in the figure, which is 108+144+108+144+180 = 684 m.
4. 

       Wich diagram shows relationship between animals, horses and lions?

Solution:

The figures have one big square, which represent the animals, and two more smaller squares, which represent the lions and the horses.

      Figure a) is wrong, because animals include lions, but lions don’t include horses.    

    Figure b) is wrong, because animals include both of lions and horses, not just only one of them.

    Figure c) is wrong, because lions and horses are different species, so there is no contact between them. 

     With all of this being said, figure d) is correct, because animals include both lions and horses, and there is no contact between the species.

The graph of the linear function

It considers the function f: R — R >, f (x) = p x + q, where p and q are real numbers.

a)     find the real numbers p and q knowing that

f (1) = 1 and f (2) =-1

b)     for p = -2 and q = 3, graph the function using xOy coordinates system. 

Geometry - an Aquarium a right prism

The figure is represented schematically an Aquarium in the form of a right prism with square floor, which features 8 dm floor side and the lateral edge of Aquarium of 5 dm. The faces of the Aquarium are made of glass. The Aquarium floor is made from a different material. Aquarium is not covered. Into Aquarium is located water up to a height of 4dm (glass thickness neglect).

a) Calculate how many litres of water are in the Aquarium,

b) Calculate how many square metres of glass are needed to into 100 pcs. of Aquarium, having the dimensions given in the hypotesis,

c) Show that, at any time, the distance between two fish is less than or equal to 12 dm.

Geometry - congruent triangles

The triangle ABC has the measure of 90 degrees of the angle A, the bisector of the angle ABC intersects in point D the side AC and DE is the altitude of the triangle BDC. Find that:

a)  AE is perpendicular to BD ; 

b)  AE is parallel to FC,  where F is the intersecting point of AB and DE: 
 {F} = AB ∩ DE.

Percents/ Rule of Three / Problem

In the figure bellow it is represented the outline of a rectangular room with an area of 48 square meters. We note ABCD this rectangle.

It is known that the width is three quarters (¾ ) of the length of the room.

Inside the room is a fireplace (a stove), represented in the figure by the square MNPD with the one meter side.

In the room it is installed hardwood floor except shaded area MNPD where is the stove.
a) find the length of the room;
b) knowing that losses represent 10% of the floor area that will be covered with parquet flooring, show that it is enough to buy 51.7 square meter of needed parquet flooring,
c) the parquet flooring is sold in packs each containing 2,5 square meters of parquet flooring. The price of each box with parquet flooring is 43 $. Find the minimum amount of money required to purchase needed parquet flooring.