Mathematics, 31.10.2019 02:31 senituliii

Two boats depart from a port located at (–10, 0) in a coordinate system measured in kilometers, and they travel in a positive x-direction. the first boat follows a path that can be modeled by a quadratic function with a vertex at (0, 5), and the second boat follows a path that can be modeled by a linear function and passes through the point (10, 4). at what point, besides the common starting location of the port, do the paths of the two boats cross?

Answers: 2

Show answers

Answers

Answer from: imsatyam9365

(6, 3.2)

Step-by-step explanation:

Stating point of both boats = (-10,0)

The first boat follows a path that can be modeled by a quadratic function with a vertex at (0, 5).

Vertex form of a quadratic function is

y=a(x-h)^2+k

where, a is constant, (h,k) is vertex.

Vertex is (0,5). Substitute h=0 and k=5 in the above equation.

y=a(x-0)^2+5

y=ax^2+5 .... (1)

It is passes through the point (-10,0). So, the equation must be satisfy by the point (-10,0).

0=a(-10)^2+5

-5=100a

$a=-\frac{5}{100}$

$a=-\frac{1}{20}$

Substitute this value in equation (1).

y=-0.05x^2+5

So, the The first boat follows a path that can be modeled by y=-0.05x^2+5 .

The second boat follows a path that can be modeled by a linear function and passes through the point (10, 4).

If a line passes through two points (x_1,y_1) and (x_2,y_2) , then the equation of line is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

The line is passes through the points (-10,0) and (10,4).

$y-0=\frac{4-0}{10-(-10)}(x-(-10))$

y=0.2(x+10)

y=0.2x+2

The second boat follows a path that can be modeled by y=0.2x+2.

Plot both functions on the graph.

From the graph it is clear that the both functions intersect each other at (-10,0) and (6,3.2).

Therefore, the paths of the two boats cross at (6,3.2).

Two boats depart from a port located at (–10, 0) in a coordinate system measured in kilometers, and

Answer from: muffin261

The point of intersection is (6, 16/5).

Step-by-step explanation:

Finding the equation of the path of the first boat.

The vertex of the parabola is at (0, 5). A point on the parabola is at (-10, 0). By symmetry, point (10, 0) is also on the parabola. We know the coordinates of three points on the parabola, so we can find the equation of the parabola.

The points are: (0, 5), (-10, 0), (10, 0)

The general form of the equation of a parabola is

y = x^2 + bx + c

We enter each of the points for x and y, and we will have a system of 3 equations that we can solve to find a, b, and c.

Point (0, 5)

y = ax^2 + bx + c

5 = a(0^2) + b(0) + c

c = 5

Point (-10, 0)

y = ax^2 + bx + c

0 = a(-10)^2 + b(-10) + 5

100a - 10b = -5 Eq. 1

Point (10, 0)

y = ax^2 + bx + c

0 = a(10)^2 + b(10) + 5

100a + 10b = -5 Eq. 2

Add Eq. 1 to Eq. 2:

200a = -10

$a = \dfrac{-10}{200}$

$a = -\dfrac{1}{20}$

Now we find b using Eq. 1.

100a - 10b = -5

$100 \times (-\dfrac{1}{20}) - 10b = -5$

-5 - 10b = -5

-10b = 0

b = 0

We have a = -1/20, b = 0, c = 5.

The quadratic equation is

$y = -\dfrac{1}{20}x^2 + 5$

We can find the linear equation of the path of the second boat by using the two given points.

y - y_1 = m(x - x_1)

where m is the slope and is

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

The points are: (-10, 0), (10, 4)

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

$m = \dfrac{0 - 4}{-10 - 10}$

$m = \dfrac{-4}{-20} = \dfrac{1}{5}$

y - y_1 = m(x - x_1)

$y - 0 = \dfrac{1}{5}(x - (-10))$

$y = \dfrac{1}{5}(x + 10)$

$y = \dfrac{1}{5}x + 2$

Now that we have the equations of the paths of the two boats, we solve them as a system of equations.

$y = -\dfrac{1}{20}x^2 + 5$

$y = \dfrac{1}{5}x + 2$

Since both expressions equal y, we set the expressions equal to each other and solve a quadratic equation.

$-\dfrac{1}{20}x^2 + 5 = \dfrac{1}{5}x + 2$

Multiply both sides by -20.

x^2 - 100 = -4x - 40

x^2 + 4x - 60 = 0

(x - 6)(x + 10) = 0

x - 6 = 0 ~or~ x + 10 = 0

x = 6~or~x = -10

We already knew one solution had to be x = -10 from the point of departure of both boats.

x = 6

$y = \dfrac{1}{5}x + 2$

$y = \dfrac{1}{5} \times 6 + 2$

$y = \dfrac{6}{5} + 2$

$y = \dfrac{6}{5} + \dfrac{10}{5}$

$y = \dfrac{16}{5}$

The point of intersection is (6, 16/5).

Answer from: Quest

This is what i found “answer: a rigid transformation preserves both the side lengths and the angle measures of a polygon. step-by-step explanation: rigid transformations are the transformations which does not affect the size and the shape of figure .” i hope this you for what your looking for. can you plz mark me brainliest?

Answer from: Quest

according to egenuity the answer is

step-by-step explanation:

d. 344.22

Another question on Mathematics

Mathematics, 21.06.2019 17:00

Acylinder and its dimensions are shown. which equation can be used to find v, the volume of the cylinder in cubic centimeters?

Answers: 1

Answer

Mathematics, 21.06.2019 20:30

Interest begins accruing the date of the transaction except for auto loans mortgages credit card cash advances credit card purchases

Answers: 1

Answer

Mathematics, 21.06.2019 23:00

Jorge wants to determine the enlarged dimensions of a digital photo to be used as wallpaper on his computer screen. the original photo was 800 pixels wide by 600 pixels high. the new photo will be 1,260 pixels wide. what will the new height be?

Answers: 1

Answer

Mathematics, 22.06.2019 02:00

An amusement park manager determined that about 23 of all customers would wait in long lines to ride the new roller coaster. which simulation could not be used to answer questions about whether a customer would wait?

Answers: 1

Answer

You know the right answer?

Two boats depart from a port located at (–10, 0) in a coordinate system measured in kilometers, and...

Questions

Mathematics, 06.06.2021 21:40

Whoever answers with the spots to graph I will like, and give the brainiest! Please help me!!!

Mathematics, 06.06.2021 21:40

Find the value of x in the isosceles triangle shown below

Mathematics, 06.06.2021 21:40

Use the diagram to answer the questions. Is line m parallel to line n? Explain.Is line m perpendicular to line k? Explain.

Mathematics, 06.06.2021 21:40