Two Tibetan monks are having a race along a straight path. They both start at the same time, and the race ends in a tie. Prove that at some time during the race the two monks have exactly the same speed.

Answers

Answer 1

This is so provided that the velocity changes continuously in which case we can apply the mean value theorem.
Velocity (v) is the derivative of displacement (x) :
v = dx/dt
Monk 1 arrives after a time t* and Monk 2 too.
Name v1(t) and v2(t) their respective velocities throughout the trajectory.
Then we know that both average velocities were equal :
avg1 = avg2
and avg = integral ( v(t) , t:0->t*) / t*
so
integral (v1(t), t:0->t*) = integral (v2(t), t:0->t*)
which is the same of saying that the covered distances after t* seconds are the same
=> integral (v1(t) - v2(t) , t:0->t*) = 0
Thus, name v#(t) = v1(t) - v2(t) , then we obtain
=> integral ( v#(t) , t:0->t*) = 0
Name the analytical integral of v#(t) = V(t) , then we have
=> V(t*) - V(0) = 0
=> V(t*) = V(0)
So there exist a c in [0, t*] so that
V'(c) = (V(t*) - V(0)) / (t* - 0) (mean value theorem)
We know that V(0) = V(t*) = 0 (covered distances equal at the start and finish), so we get
V'(c) = v#(c) = v1(c) - v2(c) = 0
=> v1(c) = v2(c)
So there exist a point c in [0, t*] so that the velocity of monk 1 equals that of monk 2.

Related Questions

A child takes a nap averaging three hours and gets an average of 12 hours of sleep at night. Nap time and night time sleep can each vary by 30 mins. What are the possible time lengths for the child's nap and night time sleep? ...?

Answers

Final answer:

The possible time lengths for the child's nap are 2.5 to 3.5 hours, and for night time sleep are 11.5 to 12.5 hours, as each can vary by up to 30 minutes.

Explanation:

The question asks about the possible time lengths for a child's nap and night time sleep, given that each can vary by 30 minutes. The average nap time is three hours, so the possible nap lengths range from 2.5 hours (3 hours - 30 minutes) to 3.5 hours (3 hours + 30 minutes). Similarly, the average night time sleep is 12 hours, meaning the possible night time sleep lengths range from 11.5 hours (12 hours - 30 minutes) to 12.5 hours (12 hours + 30 minutes).

Final answer:

The child's nap can be from 2.5 to 3.5 hours and night sleep from 11.5 to 12.5 hours, totaling a possible sleep range of 14 to 16 hours per day.

Explanation:

The child's nap time can vary by 30 minutes from the average of 3 hours, meaning the possible lengths for nap time range from 2.5 hours to 3.5 hours. Similarly, the night time sleep also has a variability of 30 minutes from the average of 12 hours, which gives a range of 11.5 hours to 12.5 hours for night time sleep. Therefore, the possible lengths for the child's total sleep time can range from a minimum of 14 hours (2.5 hours nap + 11.5 hours night) to a maximum of 16 hours (3.5 hours nap + 12.5 hours night).

A dvd costs $20 the sales tax is 6.5% how much sales tax do you have to pay

Answers

6.5% of 20....turn percent to decimal..." of " means multiply
0.065(20) = 1.30...so the sales tax is $ 1.30

what's is width of a rectangle with an area of 5/8 in and a length of ten inches

Answers

The area of a rectangle is length times width, or a = l*w. All we have to do is plug in our variables, like so: 5/8 = 10w. Divide both sides by 10 and you get a width of .0625.

why is a circle not a polygon

Answers

Because a polygon is a shape that has equal sides and no curves.

Let's first go over the definition of a polygon.

A polygon a figure formed by coplanar segments. The segments are called sides.

A circle is not considered a polygon because it's formed by a curve rather than line segments.

the table below represents the total number of laps jessica swam in the pool this summer.

week 1 18 laps week 2 28 laps week 3 38 laps week 4 48 laps

if the linear function representing the number of laps jessica swam each week is f(w)=10w+8. what does the 10 represent in this function?

Answers

for weak 1 if we implement w=1 we see that f(w) = 18

we can see that w represents number of weeks. Observing f(1) and f(2) and etc we can see that each next week number of laps is increasing by 10
f(2) - f(1) = 10
f(3) - f(2) = 10

that means that 10 represents number of laps jessica swims each week.

In the function f(w) 10w+8, the number 10 represents the number of laps Jessica increased her swim by each week.

To find out what the coefficient 10 represents in the linear function f(w) = 10w + 8, we need to understand the structure of the function.

In this function:

- w represents the week of the summer.

- f(w) represents the number of laps Jessica swam in week w.

- The coefficient 10 is multiplied by w, indicating a relationship.

Since the linear function represents the number of laps Jessica swam each week, the coefficient 10 represents the number of laps Jessica increased her swim by each week.

So, the correct answer is: C. The number of laps Jessica increased her swim by each week

Complete question:

The table below represents the total number of laps Jessica swam in the pool this summer:

Week Laps

1 18

2 28

3 38

4 48

If the linear function representing the number of laps Jessica swam each week is f(w) 10w+8, what does the 10 represent in this function?

The number of laps Jessica swam in total

The week of the summer that Jessica swam

The number of laps Jessica Increased her swim by each week

A The number of laps Jessica swam in week

the function p(t)=350e^-.2t represents the population of a city, in thousands, t years after 2010. determine the population of the city in 2014

Answers

350×e^(0.2×4)
=778.9393249724

HELP ASAP PLEASE!!!

In the past 3 years, Hannah’s savings account has earned $90 in interest. She initially deposited $600. What simple annual interest rate is the bank paying on this account?

A. 2.7%
B. 3.5%
C. 4.0%
D. 5.0%

Answers

It is D. 5.0% The formula for simple interest is I=Prt
I is interest, p is amount originally deposited, r is the rate, and t is time.

I=prt ,,90=600*3r,,,,90=1800r,,r=0.05*100,,r=5%

In 1992, jason bought a gallon of gas for $1.23. yesterday, he bought a gallon of gas for $2.19. what is the percentage increase of the price of a gallon of gas from 1992 to yesterday? if necessary, round to the nearest tenth of a percent.

Answers

1992- a gallon - 1.23dollars
yesterday - 2.19 dollars

2.19dollars - 1.23 dollars = 0.96 dollars
percentage increase is (0.96 / 1.23) x 100 = 78.0%

Answer:

39.7

Step-by-step explanation:

1/5n -3/4 + 7/10n

Need to simplify

Answers

·1/5n becomes 4/20n
·Leave 3/4 alone
·7/10n becomes 14/20n
·Combine like terms: 14/20n + 4/20n = 18/20n
·Simplify 18/20 to 9/10
·Final Answer : 9/10 - 3/4

Hope this helps you out in future problems!

A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an altitude of 700 feet.

What is the average rate of change?

A 75 feet per hour

B 200 feet per hour

C 116 feet per hour

D 100 feet per hour

Answers

I would say 75 feet per hour, though I am not 100% sure.

I first found the difference between hours (6-2 = 4) and then found the difference between distance (700 - 400 = 300). I then divided distance by hours and got 75.

Factor the expression.

9x^2 + 48x + 64

Answers

(3x+8)(3x+8) is the answer :))

Ashley's penny bank is 1/4 full. After she adds 400 pennies, it is 2/3 full. How many pennies can Ashley's bank hold?

Answers

find change
1/4 to 2/3
times 1/4 by 3/3 and 2/3 by 4/4
3/12 to 8/12
change of 5/12

5/12=400
divide both sides by 5
1/12=80
times 12 both sides
12/12=960

it can hold 960 pennies

Given that the hypotenuse of this right triangle is 25 cm long and that tan(θ) = 3find the values of x and y, accurate to the nearest tenth.

Answers

Answer:x ≈ 7.9; y ≈ 23.7

Step-by-step explanation:

The right triangle above you can use SOHCAHTOA and determine that tanθ =

. Replacing tanθ with 3 you get

3 =

3x = y

Then use the Pythagorean Theorem to

x2 + y2 = 252. Replace y with 3x

x2 + 9x2 = 625

10x2 = 625

x2 =

625

x =

= 7.9

Since,

y = 3x

y = 3(7.9) = 23.7

Write a real-world prblem in which you would need to convert pints to cups

Answers

P(pints)
C( US cups)
2P=C

what is the answer to this : (3m 3 - y)(3m 3 - y)

Answers

Good evening Shalanda

(3m³ - y)(3m³ -y)
We need to simplify this
Distribute
(3m³)(3m³)+(3m³)(-y)+(-y)(3m³)+(-y)(-y)
= 9m^6 - 3m³y - 3m³y +y²
= 9m^6 -6m³y +y²

I hope that's help !

You measure a piece of rope and find its length to be 13 m. Find the percent error of this measurement.

Answers

You cannot find the percent error of data without knowing the true length of the rope, as percent error is essentially a comparison between what you measured and what it actually is.

Which equation, when graphed, has x-intercepts at (−1, 0) and (−5, 0) and a y-intercept at (0, −30)?

f(x) = −6(x + 1)(x + 5)
f(x) = −6(x − 1)(x − 5)
f(x) = −5(x + 1)(x + 5)
f(x) = −5(x − 1)(x − 5)

Answers

The first one is correct. If you plug zero in for the x's, you will get the y intercept (since the value of x on the y axis is zero). Setting the two factors each to zero and solving you get the x intercepts -1 and -5.

Answer:

Option A

Step-by-step explanation:

The equation has x intecepts at -1 and -5.

Hence function would be of the form

a(x+1)(x+5) for some real a.

To find a, let us use y intercept

When x =0 we have y =a(1)(5)

y intercept is given as -30

i.e. 5a = -30

a=-6

f(x) = -6(x+1)(x+5)

Option A is the right answer.

For each function, what is the output of the given input?
For f(x) = 4x+7, find f(4)

Answers

For f ( x ) = 4 x + 7:
we just have to plug in 4 instead of x.
f ( 4 ) = 4 * 4 + 7 =
= 16 + 7 = 23
Answer: f ( 4 ) = 23

Answer:

f(4)=23

Step-by-step explanation:

Plug 4 in to the function

f(4)=4(4)+7

f(4)=16+7

f(4)=23

Final answer: 23

Kalon has $175 and needs to save at least $700 for a new computer. If he can save $35 per week, what is the minimum number of weeks Kalon will need to save to reach his goal?
A.10 weeks
B.12 weeks
C.15 weeks
D.20 weeks

Answers

in order to find out the answer, you just need to calculate :

(700 - 175)/35

= C. 15 weeks

hope this helps

A sample of carbon-12 has a mass of 6.00 g. How many atoms of carbon-12 are in the sample?

I have the answer but I don't understand how to solve it.

Answers

Carbon - 12 : m = 6.00 g
From the Periodic system: Molar mass of C - 12 : M = 12 g / mol
n = m / M = 6 g / 12 g/mol = 0.5 mol ( the number of moles )
N a = 6.02 * 10 ^23 mol^(-1) ( Avogadro`s constant )
N = n * N a = 0.5 mol * 6.02 * 10 ^23 = 3.01 * 10^23
Answer:
3.01 * 10^23 atoms of Carbon -12 are in the sample.

What is the amount of sales tax owed on a $39 jacket if the tax rate is 5%?

Answers

$40.95 , Total tax is $1.95

Which of the following are not congruence theorems for right triangles? Check all that apply.

A. LA
B. HL
C. AA
D. HA
E. HH
F. LL

Answers

HA means that the hypotenuse and the acute angle of two triangles are congruent. This does prove congruence.
LL means if two legs of two triangles are congruent, the two hypotenuses are congruent. This is similar to SSS congruence which proves congruence.
LA : Since we have two right triangles, three angles of the triangle and a side are congruent. By ASA, the right triangles are congruent.

Answer:

E. HH, C. AA

Step-by-step explanation:

Got these from the quiz

I know the answer, but how to solve?

Answers

The answers are:
_________________
Question # (1): t = 15.8 (when rounded to the nearest tenth).
_____________________
Question # (3): The length of Side EF is 7.5 (when rounded to the nearest tenth), and the length of side DF is 2.7 (when rounded to the nearest tenth).
_______________________
Here is how to solve for "Question #1" and "Question #3":
_________________
Let us start with "Question #1":
_________________
Remember the phrase: "SOH CAH TOA"; in which:
_________________
Sin = opposite/hypotenuse; Cos = adjacent/ hypotenuse;
Tan = opposite/adjacent.
__________________
We are given an triangle; one angle has a measurement of 44°. Considering this given angle, the respective hypotenuse given is "22"—along with a variable "t" for the respective "adjacent" side.

Using the above: "SOH CAH TOA" method; it would make sense to use the "CAH" {cosine = adjacent/hypotenuse} part of the mnemonic to solve for "t".
___________________________________________________________
Reason: We are given the angle measure 44°; the hypotenuse value ("22"); so the cosine of 44, or "cos 44" = adjacent/hypotenuse = "t" / 22 .
→ Write as: cos 44 = t / 22 ; Solve for "t" .
→ (cos 44)*22 = t ; → t = 22 * (cos 44);
→ t = 22 *(cos 44) = 22 * (0.719339800339) → Use a scientific calculator:
→ t = 15.825475607458 ; → Round to the nearest tenth, as indicated in the problem, to get:
________________________
→ t = 15.8 ; → which is the answer.
____________________________________
"Question #3: Find the missing lengths in the triangle. Round to the nearest tenth." [Note: There are 2 (two) missing lengths to find.].
___________________________
To get the length of side "EF"; let us use the "CAH" part of the mnemonic,
"SOH CAH TOA", (in the aforementioned previous problem given).

The "CAH" stands for the formula/property to calculate for the "cosine"—specifically, "cosine = adjacent/hypotenuse". We choose this one because we are given the measure of "angle E = 20° "; and we are given the hypotenuse, "side ED = 8"; and we want to solve for the adjacent side, which is "EF" (which we will call: "x").
→ So, cos 20 = adjacent/ hypotenuse = x /8; → Solve for "x""
__________________________
(cos 20)*8 = x ; → 8*(cos 20) = x ; →
→x = 8 * (cos 20) = 8 *(0.939692620786) → Use a scientific calculator: = 7.517540966288 ;→ Round to the nearest tenth, as indicated in the problem, to get:
__________________________
→ 7.5 ; which is the answer for the length of side EF.
__________
Now, we need to find the unknown length of the other side, "Side DF".
Here is one method:
_______________________
Find the measure of "Angle D". Note that for any triangle, all three sides must add up to 180°. We know that this is a right triangle, with one angle ("Angle F") having a measurement of 90°. We are also given that another angle ("Angle E") has a measurement of 20°.
→ So, the measurement of "Angle D" = 180 - (90 + 20) = 180 - 110 = 70.
→ We want to solve for the length of side "DF", which is adjacent to "Angle D" (which we know is 70°).
_____________________________________________
Let us use the "CAH" part of the mnemonic,
"SOH CAH TOA", (in the aforementioned previous problem given).
_________________________________________
The "CAH" stands for the formula/property to calculate for the "cosine"—specifically, "cosine = adjacent/hypotenuse". We choose this one because we are given the measure of angle "D"= 70°; and we are given the hypotenuse = 8. The "adjacent", or "Side DF", is the unknown value (let us make it "x").
________________________________________________
So, cos 70 = adjacent / hypotenuse = x/ 8 ; Solve for "x" :
_________________________
cos 70 = x / 8; (cos 70) * 8 = x ; 8 * (cos 70) = x;
_________________________
x = 8 * (cos 70) = 8 * (0.342020143326) → Use a scientific calculator → = 2.736161146608 ;
→Round to the nearest tenth, as indicated in the problem, to get:
________________________________________
→ 2.7 ; which is our answer, for the length of side DF.
___________________________________________
So, the answer for "Question 1" is: t = 15.8 (when rounded to the nearest tenth).
___________________________________________
For "Question #3", the length of Side EF is 7.5 (when rounded to the nearest tenth), and the length of side DF is 2.7 (when rounded to the nearest tenth).

find three solutions of the equation y=2

Answers

If y=2 that means that your slope m=0. Therefore any value you substitute for x into the equation y=mx + b will give you the value y=2. This also means that 2 is equal to b, your y-intercept. If your slope is 0 then, you have a horizontal line at y=2, so any value of (x,2) will be a solution to this problem, because your y-value will ALWAYS be 2, because 0 multplied by any number will provide 0. So subsitute your values for m and b into the slope intercept equation, and you'll see y=2, no matter what x you decide to substitute:

y= mx + b
= 0x + 2
= 2
So if I choose x to equal 3, 6, or 5 and substitute them individually into the equation for x to find y, my solutions would always give y=2.

y= mx + b
y= 0(3) +2
y= 2

Do the same with 5 and 6 and your results should be:
(3,2)
(5,2)
(6,2)

8x + 2y= 10
Solve the equation for y ...?

Answers

8x+2y=10
-8x -8x

2y=-8x+10
_ _ _
2 2 2

y=-4x+5

The solution to your problem is as follows:

8x + 2y = 10

Transpose:

2y = 10 - 8x

Divide both sides by 2:

y = 5 - 4x
I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!

ABC has vertices A(0,6) B(4,6) C(1,3). Sketch a graph of ABC and use it to find the orthocenter of ABC. Then list the steps you took to find the orthocenter including any necessary points or slopes you had to derive.

Answers

To find the orthocenter of triangle ABC with vertices A(0,6), B(4,6), and C(1,3), one must plot the triangle, calculate the slopes of its sides, find the slopes and equations of the altitudes, and determine their intersection point, which is the orthocenter.

The question involves finding the orthocenter of the triangle ABC with vertices A(0,6), B(4,6), C(1,3). To find the orthocenter, we have to identify the altitudes of the triangle and find their point of intersection. Here are the steps:

Plot the points A(0,6), B(4,6), C(1,3) on the Cartesian coordinate system and draw the triangle ABC.Find the slopes of the sides of triangle ABC: The slope of AB is 0 (horizontal line), the slope of BC and AC can be found using the formula [tex](y_2 - y1) / (x_2 - x1)[/tex].Determine the slopes of the altitudes: The slope of an altitude is the negative reciprocal of the slope of the side it is perpendicular to.Find the equations of the lines representing the altitudes by using the point-slope form of the equation of a line, which involves a point on the line (vertex of the triangle) and the slope you've just found.Calculate the point of intersection of any two altitudes. The intersection point is the orthocenter of the triangle.

Upon completing these steps, the point of intersection is the orthocenter of the triangle ABC.

Convert 1.50 μm² to square meters.

Answers

The solution to the problem is as follows:

1∙5µm² = ? m²

1m * 1m = 1m²
But 1m = 1,000,000µm = 1x10^6µm
1x10^6µm * 1x10^6µm = 1m²
1x10^12µm² = 1m² (Now, divide both sides by 1x10^12).
1x10^12µm² /(1x10^12) = 1m² /(1x10^12)
1µm² = 0∙000 000 000 001 m² (x both sides by by 1∙5).
1∙5µm² = 0∙000 000 000 0015 m²

1∙5µm² = 1∙5x10^-12 m²

I hope my answers has come to your help. God bless and have a nice day ahead!

Select the equation of the line that passes through the point (3, 5) and is perpendicular to the line x = 4.

y = 5
x = 5
y = 4
x = 3

Answers

Through the point ( 3, 5 ) passes the infinitely many lines, but only one is also perpendicular to the line x = 4.
Answer:
A ) y = 5
Graph is in the attachment.

b+c=10/a
solve for a ...?

Answers

Final answer:

To solve the equation b + c = 10/a for a, isolate a on one side of the equation by multiplying both sides by a, distribute a to both terms inside the parentheses, and then solve for a by dividing both sides by (b + c).

Explanation:

To solve the equation b + c = 10/a for a, we need to isolate a on one side of the equation. We can start by multiplying both sides of the equation by a to get rid of the denominator:

a(b + c) = 10

Next, we can distribute a to both terms inside the parentheses:

ab + ac = 10

Finally, we can solve for a by dividing both sides of the equation by (b + c):

a = 10 / (b + c)

Marge has 4 more than 2 times as many books as Susan. If Marge has 22 books and x is the number of books that Susan has, the relationship can be written as 4 + 2x = 22. How many books does Susan have?

8

9

10

11

Answers

Susan has 9 books. If the equation is 4+2x=22 Then you would first subtract 4 from 22 and get 18. Then you divide 2 from 18 and get 9, your answer. If you need me to explain it more, just ask!

First you subtract 4 from both sides to get 2x=18 then you divide both sides by two to get 9

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