The most beautiful cable-stayed bridges

The most beautiful bridges are cable-stayed. The vertical pylons are connected by a huge sagging chain. The cables that hang from the chain and support the bridge deck are called shrouds.

The figure shows a diagram of one cable-stayed bridge. Let's introduce a coordinate system: let's direct the Oy axis vertically along one of the pylons, and direct the Ox axis along the bridge deck, as shown in the figure. In this coordinate system, the line along which the bridge chain sags has the equation:

where and are measured in meters. Find the length of the cable located 100 meters from the pylon. Give your answer in meters.

The solution of the problem

This lesson demonstrates the solution of an interesting and original cable-stayed bridge problem. If this solution is used as an example for solving problems B12, preparation for the USE will become more successful and effective.

The figure clearly shows the condition of the problem. For successful solution it is necessary to understand the definitions - guy, pylon, chain. The line along which the chain sags, although it looks like a parabola, is actually a hyperbolic cosine. The given equation describes the chain slack line relative to the coordinate system. Thus, to determine the length of the cable located in meters from the pylon, the value of the equation is calculated for . In the course of calculations, one should strictly observe the order of execution of such arithmetic operations like: addition, subtraction, multiplication, exponentiation. The result of the calculation is the desired answer to the problem.

The cafe operates next rule A: 25% discount applies to the part of the order that exceeds 1000 rubles. After playing football, a student company of 20 people made an order for 3,400 rubles in a cafe. Everyone pays the same.
How many rubles will each pay?

Task 1. Option 247 Larina. USE 2019 in mathematics.

The diagram shows the average monthly air temperature in Nizhny Novgorod for each month in 1994. Months are indicated horizontally, temperatures in degrees Celsius are indicated vertically.
Determine the difference between the highest and lowest temperatures in 1994 from the diagram. Give your answer in degrees Celsius.
Decision
Task 2. Option 247 Larina. USE 2019 in mathematics.
Side isosceles triangle equals 10. From the point taken on the basis of this triangle, two straight lines are drawn, parallel to the sides.
Find the perimeter of the parallelogram bounded by these lines and the sides of the given triangle.
Decision
Task 3. Option 247 Larina. USE 2019 in mathematics.
Throw two dice.
Find the probability that the product of the rolled points is greater than or equal to 10. Round your answer to the nearest hundredth.
Decision
Task 4. Option 247 Larina. USE 2019 in mathematics.
Find the root of the equation: .
If the equation has more than one root, indicate the larger one.
Decision
Task 5. Option 247 Larina. USE 2019 in mathematics.
Find the inscribed angle based on the arc that is 1/5 of the circle.
Decision
Task 6. Option 247 Larina. USE 2019 in mathematics.
The figure shows the graph of the function y=f(x). Find among the points x1,x2,x3... those points where the derivative of the function f(x) is negative.
In response, write down the number of points found.
Decision
Task 7. Option 247 Larina. USE 2019 in mathematics.
How many times the volume of a cone circumscribed near a regular quadrangular pyramid is greater than the volume of a cone inscribed in this pyramid?
Decision
Task 8. Option 247 Larina. USE 2019 in mathematics.
Decision
Task 9. Option 247 Larina. USE 2019 in mathematics.
The figure shows a diagram of a cable-stayed bridge. The vertical pylons are connected by a sagging chain. The cables that hang from the chain and support the bridge deck are called shrouds. Let's introduce a coordinate system: we direct the Oy axis vertically along one of the pylons, and direct the Ox axis along the bridge deck, as shown in the figure. In this coordinate system, the line along which the bridge chain sags has the equation y= 0.0041x 2 -0.71x+34, where x and y are measured in meters.
Find the length of the cable located 60 meters from the pylon. Give your answer in meters.
Decision
Task 10. Option 247 Larina. USE 2019 in mathematics.
Two cars left city A for city B at the same time: the first one at a speed of 80 km/h, and the second one at a speed of 60 km/h. Half an hour later, a third car followed them.
Find the speed of the third car, if it is known that from the moment when he caught up with the second car, until the moment when he caught up with the first car, 1 hour and 15 minutes passed. Give your answer in km/h.
Decision
Task 11. Option 247 Larina. USE 2019 in mathematics.
Find the smallest value of the function on the segment
Decision
Task 12. Option 247 Larina. USE 2019 in mathematics.
a) Solve the equation
b) Indicate the roots of this equation that belong to the segment [-4pi;-5pi/2]
Decision
Task 13. Option 247 Larina. USE 2019 in mathematics.
Through the middle of the edge AC of the correct triangular pyramid SABC (S - top) planes a and b are drawn, each of which forms an angle of 300 with the plane ABC. The sections of the pyramid by these planes have a common side of length 1 lying in the face ABC, and the plane a is perpendicular to the edge SA.
A) Find the cross-sectional area of \u200b\u200bthe pyramid by plane a
B) Find the cross-sectional area of \u200b\u200bthe pyramid by plane s
Decision
Task 14. Option 247 Larina. USE 2019 in mathematics.
Solve the inequality
Decision
Task 15. Option 247 Larina. USE 2019 in mathematics.
In triangle ABC, angle C is obtuse, and point D is chosen on the continuation of AB beyond point B so that angle ACD=135°. Point D` is symmetric to point D with respect to line BC, point D is symmetric to point D`` with respect to line AC and lies on line BC. It is known that √3 ∙BC=CD'', AC=6.
A) Prove that triangle CBD is an isosceles triangle.
b) Find the area of triangle ABC

3.2.2.

vertical pylons bound huge

sagging chain. The cables that

canvas

bridge, are called shrouds.

dinat: axis OU direct vertically

Oh for example

the equation

Where X and at change

located 50 meters from the pylon.

Give your answer in meters.

3.2.3. The most beautiful bridges are cable-stayed.

vertical pylons bound huge

sagging chain. The cables that

hang from the chain and support canvas

bridge, are called shrouds.

The figure shows a diagram of one

cable-stayed bridge. Let us introduce a coordinate system

dinat: axis OU direct vertically

along one of the pylons, and the axis Oh for example

wim along the bridge deck, as shown in

figure. In this coordinate system, the chain

the equation

Where X and at change

rush in meters. Find the length of the guy

located 100 meters from the pylon.

Give your answer in meters.

4. Quadratic equations

4.1.1. (prototype 27959) In the side wall

you

is changing

tap opening,

M - initial

height of the water column

- attitude

cross-sectional areas of the crane and

tank, and g- acceleration of gravity

(consider

). After how much

seconds after opening the tap in the tank remain

a quarter of the original volume is missing

4.1.2.(28081) In the side wall of the high

honeycomb of the column of water in it, expressed in

is changing

time in seconds elapsed since

tap opening,

M - initial

height of the water column

- relatively

and tank, and g- free fall acceleration

Koryanov A.G., Nadezhkina N.V.

www.alexlarin.net

nia (consider

). After some

water weight?

4.1.3.(41369) In the side wall of the high

cylindrical tank at the very bottom

crane attached. After opening the water

starts to flow out of the tank, while you

honeycomb of the column of water in it, expressed in

is changing

time in seconds elapsed since

tap opening,

M - initial

height of the water column

- relatively

Crane Cross-Section Areas

and tank, and g- free fall acceleration

nia (consider

). After some

seconds after opening the valve in the tank

a quarter of the original

water weight?

4.2.1. (prototype 27960) In the side wall

high cylindrical tank at the very

the bottom is fixed crane. After its opening

water begins to flow out of the tank, while

is changing

elementary

M/min - constant

yannye, t

Give your answer in minutes.

4.2.2.(28097) In the side wall of the high

cylindrical tank at the very bottom

crane attached. After opening the water

starts to flow out of the tank, while you

honeycomb of the column of water in it, expressed in

is changing

elementary

M/min - by-

standing, t– time in minutes elapsed

neck from the moment the tap is opened. During

how long will the water flow out of

tank? Give your answer in minutes.

4.2.3.(41421) In the side wall of the high

cylindrical tank at the very bottom

crane attached. After opening the water

starts to flow out of the tank, while you

honeycomb of the column of water in it, expressed in

is changing

elementary

M/min - constant

yannye, t– time in minutes elapsed since

the moment the valve is opened. During some

how long will water flow out of the tank?

Give your answer in minutes.

4.3.1. (prototype

Automobile,

moving at the initial moment of time

not with speed

Started tor-

permanent

acceleration

Behind t seconds after start

braking he went the way

(m). Determine the time elapsed from

moment of the start of braking, if

it is known that during this time the car

rode 30 meters. Express your answer in seconds

4.3.2.(28147) Car moving in

Started braking from a constant

acceleration

passed the way

(m). Define-

time the car traveled 90 meters.

Express your answer in seconds.

4.3.3.(41635) Car moving in

initial moment of time with speed

Started braking from a constant

acceleration

t seconds after the start of braking

Koryanov A.G., Nadezhkina N.V. Tasks B12. Application Content Tasks

www.alexlarin.net

passed the way

(m). Define-

the time elapsed since the start

braking, if you know what it is

time the car traveled 112 meters.

Express your answer in seconds.

5. Quadratic inequalities

5.1.1. (prototype 27956) Volume dependence

demand volume q(units per month) for products

monopoly enterprise from the price p

(thousand roubles.)

given

formula

The company's revenue for

month r

Determine

highest price p, at which the month

revenue

Will be at least

240 thousand rubles Give the answer in thousand rubles.

5.1.2.(28049) The dependence of the volume of demand

acceptance-monopolist

(thousand roubles.)

given

formula

The company's revenue for

month r(in thousand rubles) is calculated according to

Determine

highest price p, at which the month

revenue

will be at least

700 thousand rubles Give the answer in thousand rubles.

5.1.3.(41311) The dependence of the volume of demand

q(units per month) for pre-

acceptance-monopolist

(thousand roubles.)

given

formula

The company's revenue for me-

a month r(in thousand rubles) is calculated according to the form-

Determine the largest

price p, at which the monthly revenue

will be at least 360 thousand rubles. From-

Vet bring in thousand rubles.

5.2.1. (prototype 27957) Height above ground

the lei of the ball tossed up changes

according to law

Where h- you-

honeycomb in meters t– time in seconds, pro-

gone from the moment of the throw. How much se-

kund the ball will be at a height not

less than three meters?

5.2.2.(28065) Height above the ground

Where h– height in met-

rah, t

children to be at a height of at least 5 meters

5.2.3.(41341) Height above the ground

the ball thrown up changes according to the law

Where h– height in met-

rah, t– time in seconds elapsed since

moment of throw. How many seconds the ball boo-

children to be at a height of at least 8 met-

5.3.1. (prototype 27958) If enough

quickly rotate a bucket of water on the wind

revolution in vertical plane, then water

will not spill out. When rotating

derka the force of water pressure on the bottom does not remain

is constant: it is maximum at

bottom point and minimum at the top.

Water will not pour out if its strength

pressure on the bottom will be positive during

all points of the trajectory except the top one,

where it can be equal to zero. To the top-

her point the pressure force, expressed in

newtons, is equal to

Where m –

mass of water in kilograms

 - speed

bucket movements in m/s, L- rope length

ki in meters, g- acceleration of free

falls (consider

). From what

at the lowest speed it is necessary to rotate the

boldly so that the water does not spill out if

the length of the rope is 40 cm? The answer is

One of the most famous bridges in the world is the Golden Gate Bridge in San Francisco. You yourself have probably seen him in American films. It is designed as follows: between two huge pylons installed on the shore, the main load-bearing chains are stretched, to which, perpendicular to the ground, beams are suspended vertically. To these beams, in turn, the bridge deck is attached. If the bridge is long, additional supports are used. In this case, the suspension bridge consists of "segments".

The figure shows a diagram of one of the segments of the bridge. Let us designate the origin of coordinates at the point of installation of the pylon, direct the Ox axis along the bridge deck, and Oy - vertically along the pylon. The distance from the pylon to the beams and between the beams is 100 meters.

Determine the length of the beam closest to the pylon if the shape of the bridge chain is given by the equation:

y=0.0061\cdot x^2-0.854\cdot x+33

in which x and y are quantities that are measured in meters. Express your answer as a number in meters.

Show Solution

Decision

The beam length is the y coordinate. According to the condition of the problem, the beam closest to the pylon is located at a distance of 100 m from it. Thus, we need to calculate the value of y at the point x = 100 . Substituting the value into the chain shape equation, we get:

y=0.0061\cdot 100^2-0.854\cdot 100+33

y=61-85.4+33

y=8.6

This means that the length of the beam closest to the pylon is 8.6 meters.

Online USE test in Mathematics 2016 Option No. 13. The test complies with the Federal State educational standards 2016. JavaScript must be enabled in your browser to pass the test. The answer is entered in a special field. The answer is an integer or a decimal, for example: 4,25 (discharge division only separated by commas). Units of measure are not written. After entering the estimated answer, click the "Check" button. In the course of the decision, you can observe the number of points scored. All scores for tasks are distributed in accordance with KIM.

PART B ACTIVITIES

Anya bought a ticket for a month and made 46 trips in a month. How many rubles did she save if a monthly ticket costs 755 rubles and a one-time trip costs 21 rubles?
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