Math New Drill (Level 2) : Transformations Problem 9 OTSUMA-2014

Water was poured into a container of Fig.1 by the rate of 40 cm³ per second. 

Fig.2 is the chart which showed a relation between time and the depth of the water since it's started pouring until it becomes filled with water. 

    

Answer the following questions. 

(1) Find the length of X of Fig.1. 

(2) Find the length of Y of Fig.1.

New Math Drill : Level 2 : Transformations Problem6 JOHOKUSAITAMA-2014

There are circle with radius 2cm centered on point O and square ABCD whose length of one side is 6cm as shown in the figure. 

Point P moves inside and on the circumference of the circle. 

(1) When point Q moves on the circumference of the square, find the length of the circumference of the figure which is made by the middle point of the line segment OQ moves.

(2) Find the area of the figure made by the middle point of the line segment AP moves. 

(3) When point Q moves on the circumference of the square, find the area of the figure made by the middle point of the line segment PQ moves.    

Math New Drill : SNJ00-0506-L2 Waiting line of products passing three machines

A certain product have to pass three machines A, B, C in turn to be made. 

The time when it takes to pass A, B, C is one minute, two minutes and four minutes respectively. 

The next product should wait in front of respective machines until the previous product finishes passing. 

Answer the following questions. 

(1) Find the number of products completed to be made 25 minutes later from machines began to move. 

(2) Find the number of products which finished passing B and are waiting in front of C three hours and six minutes later from machines began to move.

EE.8 Special digital clock

It expresses as A < B that B is larger than A.
There is a relation among the six integers A, B, C, D, E, and F called 0 < A < B < C < D < E < F and A + B + C + D = 15.

(1) How many kinds of combination of (A, B, C, D) is considered in this situation ?

(2) Furthermore, there is additional relation as A + C + E = 21, B + D + F = 42 and D + E + F = 56 to be considered further.

I made a digital clock of 24 hour display of which upper part expresses o’clock with the sum total of ● mark and lower part expresses minutes with the sum total of ● mark.

For example, the table below expressed 15:21.

Then, what time does the next table express?



(3) What does display of the digital clock express 2009 minutes after the time when it was found by (2)?
Write in the following digital clock.

EE.7 Length of a candle

There is a candle which is becoming short at a fixed rate when setting fire to the candle.
I was going to investigate in how many minutes it would be burned out when fire is set to this candle.

Since I wanted to go to a toilet suddenly 6 minutes after beginning observation, I extinguished fire once and went to a toilet.
Then, after returning from the toilet, I set fire once again 5 minutes after extinguishing fire exactly and I continued observation.

The change of the length of the candle became as the graph.
Answer the following questions.

(1) Find the length of the candle 21 minutes after setting fire first.

(2) Find the time when the length of the candle becomes 3 cm after setting fire first.

EE.6 Travel distance and railway fare

Table 1 expresses the travel distance between the stations of a certain railroad.
Moreover, the relation between the travel distance of this railroad and a fare is shown as Fig. 1.
In the graph of Fig. 1, the point indicated by ○ is not included in a graph the point indicated by ● is included in the graph.

For example, the travel distance from B station to C station is 185.8 km and the fare is 3810 yen.

The travel distance from B station to D station is 240 km and the fare is 4430 yen.
Even when travel distance exceeds 260 km, the rule of the fare of Fig. 1 is applied.
Answer the following questions.

(1) Find the travel distance from A station to D station.

(2) After Taro got on from A station to B station and finished shopping, he got on from B station to D station.
Jiro got on from A station to D station without stopping over.
How much was the sum total of the fare which Taro paid more than the fare which Jiro paid?

EE.5 Telephone rate of two companies

The telephone rate of the two telecommunications companies A and B is calculated as follows every month.

A company : Base rate : 3500 yen
Telephone rate per minute : 25 yen
However, if telephone rate is less than 500 yen, only the base rate is charged.
As for the portion beyond 500 yen, 25 yen per minute is added to the base
rate.

B company : Base rate : 5000 yen
Telephone rate per minute : 20 yen
However, if telephone rate is less than 2500 yen, only the base rate is charged.
As for the portion beyond 2500 yen, 20 yen per minute is added to the base
rate.

(1) Display the graph of the relation of the call time and telephone rate of A company.

(2) Find the call time when the difference of the rate of A company and B company is 2000 yen.

EE.4 Partition in a water tank

As shown in Fig. 1, the tank of a rectangular prism is divided into X and Y by partition of a rectangle.
Water is poured into X and Y from the water pipes A and B respectively at the same time at a fixed rate.
The graph of Fig. 2 shows the relation of the height of each water surface of X and Y and the time from the start of pouring water.
The thickness of the wall of a tank and a partition is not considered.
Answer to the following questions.

(1) Find the volume of the water which is poured out in one minute from the water pipe A.

(2) Find all the time from the start of pouring water when the difference of the height of the water surface of X and Y becomes 4 cm.

(3) Find the time from the start of pouring water when the tank is filled with water.

EE.3 One water feeding pipe and two draining pipes

Fig. 1 shows a vessel of the square pillar whose area of the bottom is 300 cm2 and depth is 50 cm.
Water pours in from A pipe and water comes out from B pipe and C pipe.
The graph of Fig. 2 expresses the relation of the depth of water and time.
Answer the following questions.

(1) Water is poured in from A pipe for the first 5 minutes with closing B pipe and C pipe.
Find the amount of the water poured per minute from A pipe.

(2) A pipe and C pipe were closed for the following 6 minutes and water was poured out from B pipe.

Furthermore, B pipe was closed for the following 3 minutes and water was poured in from A pipe and poured out from C pipe.

After that, all three pipes are opened.
Then find the time from the start when a vessel is filled with water.

EE.2 Water into three tanks

There are three kinds of tanks with equal capacity which are the rectangular prism as shown in the figure.
The side of the tank on which there is the same mark expresses the same length.
The graph expresses a relation with the depth of the water and time when the same amount of water per minute is put in three tanks.
The height of the tank represented by the graph of B is 40 cm.

(1) Find the height of the tank represented by the graph of A.

(2) Find the amount of the water put in per minute.


    

EE.1 Two points moving on a diameter

The figure is a circle centering on the point O with radius of 6 cm.
The diameters AB and CD cross right-angled.
The point P leaves A and goes back and forth between A and O at 3 cm/s without stopping.
The point Q leaves B and goes back and forth between B and O at 2 cm/s without stopping.
P and Q leave at the same time.

(1) Find the area of the triangle CPQ 5 seconds after P and Q leave.
Moreover, find the numbers of seconds after leaving when it becomes the same area at next time.

(2) Find the area of the triangle CPQ 116 seconds after P and Q leave.

(3) There is a time when the area of the portion excluding the triangle CPQ from the circle is to be 107.04 cm2.
Find the number of seconds after P and Q leave when it becomes this area at 6th times.
Pi is assumed to be 3.14.

EEE.4 Taxi fare of two types of cars

As for the meter of the taxi fare, until 2000 m from starting point, it is 450 yen by a downsized car and 470 yen by a standard car.

When it exceeds 2000 m, 70 yen is added to the fare in both size of car and it is changed to 520 yen until 2400 m by a downsized car and to 540 yen until 2370 m by a standard car.

As is in the same as above, in every time the distance exceeds 400 m in case of downsized car and 370 m in case of standard car, 70 yen is added to the taxi fare.

Answer the following questions.

(1) Find the longest distance where you take a taxi and can go within 1000 yen.

Find it in each case of downsized car and standard car.

(2) When you go to the some place of 5000 m far by taxi, find the difference of the fare by the case you take downsized car and standard car.

(3) I took a taxi from A point to B point.

The number of times that fare meter was added was three in both cases that I took downsized car and standard car.

Find the range of the distance from A point to B point.

(4) I took a taxi from C point to D point.

The number of times that fare meter was added was same in both cases that I took downsized car and standard car.

Find the longest distance in the distance considered between C point and D point.

EEE.3 Change of gas rate in several months

The gas rate of every month of a certain gas company is the sum total of the fixed amount of basic charge and the charge corresponding to the amount of the gas used. 

The charge corresponding to the amount used is calculated from the unit price per m³ provided in stages corresponding to the amount used. 

The unit price is as follows.

The amount used Up to 10 m³                                160 yen per 1 m³
                               Over 10 m³ up to 30 m³             X    yen per 1m³
                               Over 30 m³                                  240 yen per 1 m³

In addition, the gas rate is calculated by 1 m³ increments. 

For example, when it is used 40 m³, the charge is calculated as (basic charge) + 10 × 160 yen + 20 × X yen + 10 × 240 yen. 

At Taro’s house, 28 m³ gas was used in October and the charge was 6,310 yen. 

Moreover, 40 m³ gas was used in December and the charge was 9,150 yen. 

Answer the following questions.

(1) Find the basic charge of gas and the unit price of X yen of a secondary stage.

(2) As for Taro’s house, the least amount of the gas used in a year is in August and the most is in February. 

In February the amount of gas used was exactly 3 times in August. 

Moreover, the gas rate in February also became exactly 3 times in August. 

Find the amount of gas used and the gas rate in February. 

EEE.2 Two points moving in a square

There is a square ABCD whose length of one side is 10 cm. E, F, G, and H of the figure are the middle points of square side, respectively.

The point O is the intersection of EF and GH.

From the point A, the point P starts from point A at 1 cm/s and moves on the line of the figure as A - E - O - F - C - G - O - H - A.

Moreover, the point Q leaves the point A at the same time with the point P and turns around the circumference of the square ABCD clockwise with fixed speed quicker than the point P.

In this case, answer the following questions.

(1) Draw the graph of the situation of the point P while it is moving on the circumference of the square ABCD.

(You must not draw the graph while the point P is not moving on the circumference of the square ABCD.)

(2) After leaving, the point Q overlapped point P for the first time at the point of 1 cm from C on CF and this overlapping was in the 2nd round.

Find the speed of the point Q.

Moreover, find the point where P and Q overlaps at the 2nd time after starting.

Write the suitable number or character in underline parts.

The point is located at __X___ cm from ___Y___ on the side ___Z____.

EEE.1 Three points moving on a rectangle

As shown in Fig. 1, there is a rectangle ABCD whose length of side AB is 18 cm and length of side AD is 42 cm.

The point P leaves A and moves at the speed of 3 cm/s clockwise rotation on the side of rectangle ABCD as A -> D -> C -> B -> A.

It will stop when it arrives at A again.

The point Q leaves B at the same time as P leaves A.

It moves counterclockwise until P stops at the speed of 2 cm/s on the side of rectangle.

Answer the following questions.


(1) In how many seconds after started is it that P and Q meet?

(2) The point R leaves A at the same time with P and moves counterclockwise on the side of the rectangle at the speed of 2 cm/s.
It moves until P stops.
The figure which connected three point P, Q, and R is considered.

(2)-1 A triangle is not made several times while three points are moving.

Find the time altogether when the triangle is not be made.

The time should be found the time since they begin to move.

For example, When the triangle is not be made all the time from x-second to y-second, you may answer as x~y seconds.

(2)-2 As shown in Fig. 2, the intersection of the line AC and the line BD is set to E.

Find the time altogether when E is on the inside and circumference of the triangle PQR.

In accordance with (2)-1, answer the time altogether.