A heavy rope, 50 ft long, weighs 0.6 lb/ft and hangs over the edge of a building 120 ft high. Approximate the required work by a Riemann sum, then express the work as an integral and evaluate it.
Exercise (a)
How much work W is done in pulling the rope to the top of the building?
Exercise (b)
How much work W is done in pulling half the rope to the top of the building?
Answer:
Exercise (a)
The work done in pulling the rope to the top of the building is 750 lb·ft
Exercise (b)
The work done in pulling half the rope to the top of the building is 562.5 lb·ft
Step-by-step explanation:
Exercise (a)
The given parameters of the rope are;
The length of the rope = 50 ft.
The weight of the rope = 0.6 lb/ft.
The height of the building = 120 ft.
We have;
The work done in pulling a piece of the upper portion, ΔW₁ is given as follows;
ΔW₁ = 0.6Δx·x
The work done for the second half, ΔW₂, is given as follows;
ΔW₂ = 0.6Δx·x + 25×0.6 × 25 = 0.6Δx·x + 375
The total work done, W = W₁ + W₂ = 0.6Δx·x + 0.6Δx·x + 375
∴ We have;
W = [tex]2 \times \int\limits^{25}_0 {0.6 \cdot x} \, dx + 375= 2 \times \left[0.6 \cdot \dfrac{x^2}{2} \right]^{25}_0 + 375 = 750[/tex]
The work done in pulling the rope to the top of the building, W = 750 lb·ft
Exercise (b)
The work done in pulling half the rope is given by W₂ as follows;
[tex]W_2 = \int\limits^{25}_0 {0.6 \cdot x} \, dx + 375= \left[0.6 \cdot \dfrac{x^2}{2} \right]^{25}_0 + 375 = 562.5[/tex]
The work done in pulling half the rope, W₂ = 562.5 lb·ft
A bag of 4 balls weighs 6 pounds. Each ball weighs the same amount. What is the weight of each ball?
Calculate the volume of this triangular prism?
Answer:
960 sq cm
Step-by-step explanation:
(16*12*10)/2
may kapangyarihan bilang pinakamataas na hukuman sa kolonya
Answer:
which language is this .
abc~def what sequence of transformation will move ABC onto def
Answer:
c
Step-by-step explanation:
Answer:
is C guys
Step-by-step explanation:
Please help me with this :)
[tex]hope \: it \: helps[/tex]
At art class, Mr. Gibson spends 5 minutes giving instructions, 35 minutes working on the project, and 15 minutes cleaning up. If class starts at 1:15 PM, what time will it end? PM
Answer:
2:10 PM
Step-by-step explanation:
5+35+15=55 mins
1:15+55 mins=2:10PM
Can Someone please help me !!! ( you have to fill out the blanks )
Answer:
1. 5(a-6) = 5a-30
2. 2(3a-b) = 6a-2b
3. 5(-5x+3) = -25x+15
4. -(2c+3) = -2c-3
Both of their first step are correct (Show your calculations to explain)
x-9 = 5
x = 14
2x-18 = 10
2x = 28
x = 14
Step-by-step explanation:
All Above
Assume v is a vector in R^5. You wish to compute a vector w such that the th entry of w, denoted wi, is the sum of vi-1,vi, vi+1. For example wi is the sum of ,vo,v1 and v2. For elements at the endpoints of w, you will only account for two elements in your sum. For example wo, (assuming zero-based indexing) is the sum of vo and v1. Construct a matrix A such that Av=w.
Answer:
Use the drawing tools to form the correct answer on the graph.
Graph the line that represents this equation:
Step-by-step explanation:
Find y' if y= In (x2 +6)^3/2
y'=
Answer:
[tex]\displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
FactoringCalculus
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
ln Derivative: [tex]\displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle y = ln(x^2 + 6)^{\frac{3}{2}}[/tex]
Step 2: Differentiate
[Derivative] Chain Rule: [tex]\displaystyle y' = \frac{d}{dx}[ln(x^2 + 6)^{\frac{3}{2}}] \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6][/tex][Derivative] Chain Rule [Basic Power Rule]: [tex]\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{3}{2} - 1} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6][/tex][Derivative] Simplify: [tex]\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6][/tex][Derivative] ln Derivative: [tex]\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot \frac{d}{dx}[x^2 + 6][/tex][Derivative] Basic Power Rule: [tex]\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2 \cdot x^{2 - 1} + 0)[/tex][Derivative] Simplify: [tex]\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2x)[/tex][Derivative] Multiply: [tex]\displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2} \cdot \frac{1}{x^2 + 6} \cdot (2x)[/tex][Derivative] Multiply: [tex]\displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)} \cdot (2x)[/tex][Derivative] Multiply: [tex]\displaystyle y' = \frac{3(2x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}[/tex][Derivative] Multiply: [tex]\displaystyle y' = \frac{6xln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}[/tex][Derivative] Factor: [tex]\displaystyle y' = \frac{2(3x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}[/tex][Derivative] Simplify: [tex]\displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}[/tex]Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives
Book: College Calculus 10e
A brochure claims that the average maximum height a certain type of plant is 0.7 m. A gardener suspects that this estimate is not accurate locally due to soil conditions. A random sample of 43 plants is taken. The mean height of the plants in the sample is 0.65m. Using a 1% level of significance, perform a hypothesis test to determine whether the population mean is different from 0.7m. Assume that the population standard deviation is 0.2 m.
Answer:
We accept the null hypothesis, that is, that the population mean is not different from 0.7.
Step-by-step explanation:
A brochure claims that the average maximum height a certain type of plant is 0.7 m.
This means that the null hypothesis is:
[tex]H_{0}: \mu = 0.7[/tex]
A gardener suspects that this estimate is not accurate locally due to soil conditions.
Hypothesis test to determine whether the population mean is different from 0.7m, which means that the alternate hypothesis is:
[tex]H_{a}: \mu \neq 0.7[/tex]
The test statistic is:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.
0.7 is tested at the null hypothesis:
This means that [tex]\mu = 0.7[/tex]
A random sample of 43 plants is taken. The mean height of the plants in the sample is 0.65m.
This means, respectively, that [tex]n = 43, X = 0.65[/tex]
Assume that the population standard deviation is 0.2 m.
This means that [tex]\sigma = 0.2[/tex]
Value of the test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{0.65 - 0.7}{\frac{0.2}{\sqrt{43}}}[/tex]
[tex]z = -1.64[/tex]
Pvalue of the test:
Since we are testing that the mean is different of a value, and z is negative, the pvalue of the test is 2 multiplied by the pvalue of z = -1.64
z = -1.64 has a pvalue of 0.0505
2*0.0505 = 0.101
Using a 1% level of significance, perform a hypothesis test to determine whether the population mean is different from 0.7m.
0.101 > 0.01, which means that we accept the null hypothesis, that is, that the population mean is not different from 0.7.
Fill in the blank so that the ordered pair is a solution of y = 22 – 9x.
( ,4)
Out of all of the topics covered so far in the course (whole numbers, fraction notation: multiplication, division, and mixed numerals), which did you enjoy the most? Which do you enjoy the least? Why?
Answer: I would say I've liked multiplication the most and division the least! I have always gotten multiplication because.. I don't really know. It's just easy for me. I can't really explain. For multiplication though.. I have never gotten it. It is the exact opposite. I've never understand how you have to skip a digit because you can't divide it to the other number.
Step-by-step explanation:
Hope this helps! This is my opinion.
I made this don't copy my answer
Which graph best represents the equation -4x + 5y = 8? A. B. C. D.
Answer:
It will look like this
Step-by-step explanation:
The equation of line y = 4 / 5x + 8 / 5 can be represented on the basis of coordinates (0, 8/5), (1, 12/5), and (10, 48/5) in the graph attached below.
What is the equation?A formula known as an equation uses the equals sign to denote the equality of two expressions.
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Given:
-4x + 5y = 8
Write the line in slope intercept form as shown below,
5y = 4x + 8
y = 4 / 5x + 8 / 5
Put the random values of x and find y,
y = 4 / 5 × 0 + 8 / 5 (when x = 0)
y = 8 / 5
y = 4 / 5 × 1 + 8 / 5 (When x = 1)
y = 12 / 5
y = 4 / 5 × 10 + 8 / 5 (When x = 10)
y = 48 / 5
Plot the coordinates (0, 8/5), (1, 12/5), and (10, 48/5) in the graph and mark the points, joining all the points to get the line.
To know more about equation:
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find the slope
!!!!!please help QUICK!!!!
Answer:
3/4
Step-by-step explanation:
We can find the slope using two points
( 0,-5) and ( 4,-2)
The slope is given by
m = ( y2-y1)/(x2-x1)
= ( -2- -5)/( 4-0)
= (-2+5)/(4-0)
=3/4
Write an emergency that could happen that would have a financial cost
Answer:
Your house burning down.
Step-by-step explanation:
You would have to pay to repair it.
Answer:
Earthquake will make the building crash and people will lost their things and money. And the building need to rebuild.
The building fire will make people lost their things and the building need to rebuild.
Tsunami will make the things near beach all destroy, all the things will lost and more people will die in disaster. All the building need to rebuild in this disaster.
Bank robbery will lost money, sometimes people will kill by the gun of the thief, and the building do not need to rebuild, maybe the window need to rebuild.
Rob the shop will make the boss of the shop lose all the money in the store, the door or window might need to rebuild.
Blasting vehicle will make the things in the car lose. the car need to put the windows on again.
Terrorist attacks will lost lives more than lost money or other things.
Volcanic eruptions will make the building near burn, and so plant and many other things. The building might need to rebuild.
That is all I can think of.
Which fraction is marked by the point on the number line?
Answer:
where is the numer line?
Step-by-step explanation:
A bucket has 16 tennis balls and 10 ping pong balls . What is the probability of selecting a ping pong ball , replacing it and then selecting a tennis ball
Answer:
40/169
Step-by-step explanation:
P(pp,tennis) = [tex]\frac{10}{26}[/tex] × [tex]\frac{16}{26}[/tex] = 40/169 in simplified form
The probability of selecting a ping pong ball, replacing it and then selecting a tennis ball is 40/169.
What is Probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
We have,
Tennis ball = 16
Ping pong balls = 10
So, probability of selecting a ping pong ball, replacing it and then selecting a tennis ball
= 10/ 26 x 16/26
= 40/169
Learn more about Probability here:
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Margo can purchase tile at a store for $0.59 per tile and rent a tile saw for $49. At another store she can borrow the tile saw for free if she buys tiles there for $1.29 per tile. How many tiles must she buy for the cost to be the same at both stores?
Margo must buy
tiles for the cost to be the same at both stores.
Answer:
Also need help
Function Notation
If f(x) = -3x2 + 2x - 8, find each of the following:
f(-1)
f(0)
f(3)
f(1/3)
PLEAS HELP THERE IS PIC drag the tiles to the correct boxes to complete the pairs. how many solutions does each equations have ?
Answer:
|2x+1|=0 has one solutions
2|x+1|=-2 has no solution
|x+21|=2 has two solution
Step-by-step explanation:
Please help I don’t gat this
Answer:
30 yards
Step-by-step explanation:
the dotted line causes a rectangle and a triangle to be formed. the formula for area of a triangle is base times height divided by 2. you subtract 6 from 9 which is 3 and multiply it by 4 as it is the height of the triangle after, divide by 2 . since 4 and 3 is 12, dividing 12 by 2 is 6 which is the area of the triangle. the area of the rectangle is 6 times 4 which is 24. add 24 and 6 and u get 30
Help! Will give brainliest to first correct answer.
Answer:
a) 7n
b) n+7
c) n -7
d). [tex]\frac{n}{7}[/tex]
Step-by-step explanation:
All you have to do for this question is look at the key words. These will help you find the correct answers.
Product = multiplication (x)Sum = addition (+)Difference = subtraction (-)Quotient = DivisionWhich statement best describes the function?
O A. The function is increasing when x < 0.
B. The function is decreasing when x > 0.
C. The function is never increasing.
D. The function is always increasing.
Answer:
D. The function is always ingcreasing
Solve for x if 26 = x/2
Answer:
x=52
Step-by-step explanation:
please help!!!!!!!!!!!!!!!!!!!!
Answer:
I am not sure but think that the answer is 1/20
I believe you convert the whole number into a mixed fraction.
5 converted into quarters is 20/4. Then you divided 1/4 by 20/4.
1/4 divided by 20/4 is 1/20 or 0.05.
Please, don't refrain to tell me if this is incorrect. Thank you.
Find the length of the third side. If necessary,
round
to
the nearest tenth.
4
7
Answer:
The answer would be 8.1
Step-by-step explanation:
The hanger image below represents a balanced equation. Find the value of n that makes the equation true.
Answer:
6
Step-by-step explanation:
16=n +10
subtract 10 from both sides and your left with 6=n
ASAP I WILL GIVE 100 POINTS AND BRAINIEST ASAP
Find the length of the model train.
Answer:
It would be 7.5 inches.
Hope this helped!!
Item 5
Which measure of central tendency best describes the data?
47, 47, 48, 54, 102, 45, 100, 48, 83
either mean or median
median only
mean only
Answer:
the answer is either mean or median because you can find both of them given the data