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4 years ago

A runner is 7/8 mile from the finish line. If she can travel 3/8 mile per minute, how long will it take her to finish the race?

Mathematics

1 answer:

luda_lava [24]4 years ago

4 0

Answer:

7/3 minutes

Step-by-step explanation:

The speed if the runner is given by:

$speed = \frac{distance}{time}$

We have that: speed =3/8 miles per minute and distance =7/8 miles

We substitute into the formula to get:

$\frac{3}{8} = \frac{ \frac{7}{8} }{time}$

$time = \frac{ \frac{7}{8} }{ \frac{3}{8} }$

$time = \frac{7}{8} \times \frac{8}{3}$

This simplifies to

$\frac{7}{3}$

Therefore it will take 7/3 miniutes to finish the race.

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Find all solutions to the equation in the interval [0, 2π).<br><br> cos 4x - cos 2x = 0

Andrew [12]

Solve cos(4x)-cos(2x)=0 ∀ 0<=x<=2pi ..............(0)

Normal solution:
1. use the double angle formula to decompose, and recall cos^2(x)+sin^2(x)=1
cos(4x)=cos^2(2x)-sin^2(2x)=2cos^2(2x)-1 .................(1)
2. substitute (1) in (0)
2cos^2(2x)-1-cos(2x)=0
3. substitute u=cos(2x)
2u^2-u-1=0
4. Solve for x
factor
(u-1)(u+1/2)=0
=> u=1 or u=-1/2
However, since cos(x) is an even function, so solutions to
{cos(2x)=1, cos(-2x)=1, cos(2x)=-1/2 and cos(-2x)} ...........(2)
are all solutions.
5. The cosine function is symmetrical about pi, therefore
cos(-2x)=cos(2*pi-2x),
solution (2) above becomes
{cos(2x)=1, cos(2pi-2x)=1, cos(2x)=-1/2, cos(2pi-2x)=-1/2}
6. Solve each case
cos(2x)=1 => x=0
cos(2pi-2x)=1 => cos(2pi-0)=1 => x=pi
cos(2x)=-1/2 => 2x=2pi/3 or 2x=4pi/3 => x=pi/3 or 2pi/3
cos(2pi-2x)=-1/2 => 2pi-2x=2pi/3 or 2pi-2x=4pi/3 => x=2pi/3 or x=4pi/3
Summing up,
x={0,pi/3, 2pi/3, pi, 4pi/3}

3 0

3 years ago

Find the slope of the line graph

Tamiku [17]

Answer:

-1

Step-by-step explanation:

the "slope" of a graph is rise / run

you divide the "rise" from one point on a graph to another by the "run" from one point to another.

We can use any two points on a graph to find this value. (There are other ways to find slope, but this is the easiest method for find the slope of a graphed line)

We can see two points on the line already pointed out, but we can also look to see other points (that require less math)

[remember, points are written as (x,y) ]
We can see a point at (1,0) and another point at (0, 1)

the "rise" between two points is the difference in y-values.

0 - 1 = -1

{you can also look at the graph/count the boxes}

So, the rise between these two points is -1.

(really, it's a drop)

the "run" between these two points is the difference in x-values.

1 - 0 = 1

so, the run between these two points is 1.

Now, all that's left to do is divide the rise / run:

-1 / 1 = -1

So, the slope of this line is -1.

this can be summed up in the following formula:

$m = \frac{y_{2} - y_{1}} {x_{2}-x_{1}}$

where $m$ is the slope of the graph,

$x_{1}$ = x-value of first point on graph

$y_{1}$ = y-value of first point on graph

$x_{2}$ = x-value of second point on graph

$y_{2}$ = y-value of second point on graph

the equation of a line

y = mx + b

incorporates the slope (as $m$ )

4 0

2 years ago

A student is using the elimination method to solve the system of equations below. What is the best first

vlada-n [284]

Answer:

The best first step would be to multiply the second equation by -2

Step-by-step explanation:

The best first step would be to multiply the second equation by -2

then you would have the following

$\ \ \ \ \ \ 4x - 5y = 2 \\-2*(2x)+ (-2)*y = (-2)*3$

and when you multiply it is easy to eliminate because you will get

$4x - 5y = 2 \\-4x -2y = 6$

and if you sum the equations you get

-7y = 8

so that is a single variable equation which is easier to solve.

7 0

3 years ago

The graph of f(x)=sin(x) is transformed into a new function, g(x) , by stretching it vertically by a factor of 4 and shifting it

Snowcat [4.5K]

Answer:

g(x) = 4 sin(x - 3)

Step-by-step explanation:

* Lets revise some transformation

- If the function f(x) translated horizontally to the right by h units, then

the new function g(x) = f(x - h)

- If the function f(x) translated horizontally to the left by h units, then

the new function g(x) = f(x + h)

- If the function f(x) translated vertically up by k units, then the new

function g(x) = f(x) + k

- If the function f(x) translated vertically down by k units, then the

new function g(x) = f(x) – k

- A vertical stretching is the stretching of the graph away from the

x-axis

- If k > 1, the graph of y = k • f(x) is the graph of f(x) vertically

stretched by multiplying each of its y-coordinates by k.

- A vertical compression is the squeezing of the graph toward

the x-axis.

- If 0 < k < 1 (a fraction), the graph of y = k • f(x) is the graph of f(x)

vertically compressed by multiplying each of its y-coordinates by k.

* Lets solve the problem

∵ f(x) = sin(x)

∵ f(x) is stretching vertically by a factor of 4

∴ f(x) multiplied by 4

∵ f(x) is shifting 3 units to the right

∴ We will change x to (x - 3)

∵ The new function is g(x)

∴ g(x) = 4 sin(x - 3)

- The attached graph for more understand

f(x) is the red graph

g(x) is the blue graph

5 0

4 years ago

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Brianna earns $5 each week for walking her neighbor's dog.

snow_tiger [21]

Answer: 35 Dollars

Step-by-step explanation: If brianna earns $5 each week for walking her neighbor's dog and she does it for 7 weeks, you have to multiply 5 by 7 to figure out how much she earns.

5 0

4 years ago

Read 2 more answers