1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
g100num [7]
3 years ago
14

The area of a sector of a circle with a central angle of 170° is 70 m 2. find the radius of the circle. (round your answer to on

e decimal place.)
Mathematics
2 answers:
Morgarella [4.7K]3 years ago
8 0
The area of a sector of a circle is A= \frac{ \theta }{360} * \pi r^2.  We have everything but the radius to fill in there, so let's do it: 70= \frac{170}{360} * \pi r^2.  To make it easy, cross out the last 0 in both the 170 and the 360, and then multiply both sides by 36 to get 2520=17 \pi r^2.  Divide both sides by 17pi and you have 47.1847596=r^2 and r = 6.9
Lesechka [4]3 years ago
3 0
Area of a circle = πr²

Area of a sector = πr²*(α/360⁰)

70 = πr² * 170/360

r² = (70*360)/(170*π)

r = √(70*360)/(170*π) ≈ 6.9 m

r ≈ 6.9 m

You might be interested in
Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
zvonat [6]

Answer:

The maximum volume of a box inscribed in a sphere of radius r is a cube with volume \frac{8r^3}{3\sqrt{3}}.

Step-by-step explanation:

This is an optimization problem; that means that given the constraints on the problem, the answer must be found without assuming any shape of the box. That feat is made through the power of derivatives, in which all possible shapes are analyzed in its equation and the biggest -or smallest, given the case- answer is obtained. Now, 'common sense' tells us that the shape that can contain more volume is a symmetrical one, that is, a cube. In this case common sense is correct, and the assumption can save lots of calculations, however, mathematics has also shown us that sometimes 'common sense' fails us and the answer can be quite unintuitive. Therefore, it is best not to assume any shape, and that's how it will be solved here.

The first step of solving a mathematics problem (after understanding the problem, of course) is to write down the known information and variables, and make a picture if possible.

The equation of a sphere of radius r is x^2 + y^2 + z^2=r^2. Where x, y and z are the distances from the center of the sphere to any of its points in the border. Notice that this is the three-dimensional version of Pythagoras' theorem, and it means that a sphere is the collection of coordinates in which the equation holds for a given radius, and that you can treat this spherical problem in cartesian coordinates.

A box that touches its corners with the sphere with arbitrary side lenghts is drawn, and the distances from the center of the sphere -which is also the center of the box- to each cartesian axis are named x, y and z; then, the complete sides of the box are measured  2x,  2y and 2z. The volume V of any rectangular box is given by the product of its sides, that is, V=2x\cdot 2y\cdot 2z=8xyz.

Those are the two equations that bound the problem. The idea is to optimize V in terms of r, therefore the radius of the sphere must be introduced into the equation of the volumen of the box so that both variables are correlated. From the equation of the sphere one of the variables is isolated: z^2=r^2-x^2 - y^2\quad \Rightarrow z= \sqrt{r^2-x^2 - y^2}, so it can be replaced into the other: V=8xy\sqrt{r^2-x^2 - y^2}.

But there are still two coordinate variables that are not fixed and cannot be replaced or assumed. This is the point in which optimization kicks in through derivatives. In this case, we have a cube in which every cartesian coordinate is independent from each other, so a partial derivative is applied to each coordinate independently, and then the answer from both coordiantes is merged into a single equation and it will hopefully solve the problem.

The x coordinate is treated first: \frac{\partial V}{\partial x} =\frac{\partial 8xy\sqrt{r^2-x^2 - y^2}}{\partial x}, in a partial derivative the other variable(s) is(are) treated as constant(s), therefore the product rule is applied: \frac{\partial V}{\partial x} = 8y\sqrt{r^2-x^2 - y^2}  + 8xy \frac{(r^2-x^2 - y^2)^{-1/2}}{2} (-2x) (careful with the chain rule) and now the expression is reorganized so that a common denominator is found \frac{\partial V)}{\partial x} = \frac{8y(r^2-x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}}  - \frac{8x^2y }{\sqrt{r^2-x^2 - y^2}} = \frac{8y(r^2-2x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}}.

Since it cannot be simplified any further it is left like that and it is proceed to optimize the other variable, the coordinate y. The process is symmetrical due to the equivalence of both terms in the volume equation. Thus, \frac{\partial V}{\partial y} = \frac{8x(r^2-x^2 - 2y^2)}{\sqrt{r^2-x^2 - y^2}}.

The final step is to set both partial derivatives equal to zero, and that represents the value for x and y which sets the volume V to its maximum possible value.

\frac{\partial V}{\partial x} = \frac{8y(r^2-2x^2 - y^2)}{\sqrt{r^2-x^2 - y^2}} =0 \quad\Rightarrow r^2-2x^2 - y^2=0 so that the non-trivial answer is selected, then r^2=2x^2+ y^2. Similarly, from the other variable it is obtained that r^2=x^2+2 y^2. The last equation is multiplied by two and then it is substracted from the first, r^2=3 y^2\therefore y=\frac{r}{\sqrt{3}}. Similarly, x=\frac{r}{\sqrt{3}}.

Steps must be retraced to the volume equation V=8xy\sqrt{r^2-x^2 - y^2}=8\frac{r}{\sqrt{3}}\frac{r}{\sqrt{3}}\sqrt{r^2-\left(\frac{r}{\sqrt{3}}\right)^2 - \left(\frac{r}{\sqrt{3}}\right)^2}=8\frac{r^2}{3}\sqrt{r^2-\frac{r^2}{3} - \frac{r^2}{3}} =8\frac{r^2}{3}\sqrt{\frac{r^2}{3}}=8\frac{r^3}{3\sqrt{3}}.

6 0
3 years ago
MA = (4x - 2) and mzB = (11x + 17)°. Find x if the angles are
charle [14.2K]
The answer is a.) 165 sorry if that’s wrong but i hope it’s right
3 0
3 years ago
Read 2 more answers
In your own words, explain why solving 4=2x uses the Division Property of Equality. Provide an example with your explanation.
balandron [24]
Solving 4=2x is using the division property because in order to get the last portion of that answer u have to divide both sides by 2x to get x by itself
For example
4=2x to get x by itself you have to divide both sides by 2x
And your final answer would be
x is equal to 2 (x=2)
4 0
3 years ago
If f (x) = -2x2 + 6, find f (-3).
Margarita [4]

Answer:

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
Kyle leaves a 15% tip for lunch bill of $12 what amount tip does Kyle leave
Aleonysh [2.5K]
Kyle leaves 1.80 because you have to do 0.15*12
8 0
3 years ago
Read 2 more answers
Other questions:
  • What number is 8.2% of 50?
    9·2 answers
  • A standard working day is 8 hours. If you were to work 125% of a normal day, how many total hours would you work?
    9·2 answers
  • Please help!!! Thank you!
    13·1 answer
  • 9) 15k + 20 – 10k = 10
    7·1 answer
  • What is 2/3+3/4+5/8 please help
    12·1 answer
  • What are the odds against selecting a red marble
    14·1 answer
  • Alguien que hable ESPAÑOL :) ​
    14·2 answers
  • Which of the following is a solution to the inequality 5x-25>18
    10·1 answer
  • Help 100 Points + brainlest math
    9·2 answers
  • You roll a 6-sided die.<br> What is P(prime)?<br><br> Write your answer as a percentage.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!