Answer:
Expected Payoff ⇒ $ 1.50 ; Type in 1.50
Step-by-step explanation:
Considering that 1 out of the 100 tickets will have a probability of winning a 150 dollar prize, take a proportionality into account;
\(100 - Number of Tickets,\\1 - Number of Tickets You Can Enter,\\1 / 100 - Probability of Winning,\\$ 150 - Money Won,\\Proportionality - 1 / 100 = x / 150, where x - " Expected Payoff "\\\\1 / 100 = x / 150,\\100 * x = 150,\\\\Conclusion ; x = 1.5 dollars\)
Thus, Solution ; Expected Payoff ⇒ $ 1.50
A cyclist is riding on a bike trail, pedaling at a speed of 20 miles per hour. the cyclist plans to pedal at this speed for the next 35 miles, plus or minus 10 miles. find the minimum and maximum number of hours the cyclist will pedal at that speed.
The cyclist will pedal at given speed for minimum 1.25 and maximum 2.25 hours.
The relation between speed, distance and time is as follows -
Speed = distance ÷ time. The maximum and minimum number of hours will depend on the distance. The maximum number of hours will be spent when maximum distance is covered, and minimum number of hours will be spent when minimum distance is covered.
Maximum distance = 35 + 10
Maximum distance = 45 miles
Minimum distance = 35 - 10
Minimum distance = 25 miles
Maximum number of hours = 45 ÷ 20
Maximum number of hours = 2.25 hours
Minimum number of hours = 25 ÷ 20
Minimum number of hours = 1.25 hours
Thus, cyclist will pedal at given speed for minimum 1.25 and maximum 2.25 hours.
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4 The scatterplot shows the relationship between the weight
pounds and the age in weeks of a certain dog breed.
Dog Weight
100
(spunod) 14E
90
80
70
60
●
●
.. how would you explain this answer
Based on the given scatterplot the best prediction of weight of the 28 week old dog will be approximately 75 lb pounds.
In the given scatterplot in the x-axis we find the age of the dog in weeks and on the y-axis there is the weight of the dog per each week.
From the given scatterplot after observing closely, we can see that at 28 weeks which is in between 26 and 30, the weight of the dog will be approximately present at 75 lb which is in between 70 lb and 80 lb pounds on the y-axis.
From the above explanation, we can conclude that the weight of the 28 week old dog is 75 lb pounds.
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Given question is not having enough information, so I am attaching the complete question below:
how to find the least common denominator of rational expressions
The least common denominator (LCD) for the rational expressions A and B is (x - 2)(x + 2).
To find the least common denominator (LCD) of rational expressions, you can follow these steps:
Identify the denominators of the rational expressions involved.
Factorize each denominator completely.
Identify the common factors among all the denominators. Include each factor the maximum number of times it appears in any of the denominators.
Multiply together all the common factors identified in step 3. This product represents the LCD.
Simplify the LCD if possible by canceling out any common factors that appear in both the numerator and denominator of any rational expression.
Here's an example to illustrate the process:
Consider the rational expressions:
A = (3x) / (x² - 4)
B = (2) / (x - 2)
Step 1: Identify the denominators
Denominator of A: x² - 4
Denominator of B: x - 2
Step 2: Factorize each denominator
x² - 4 = (x - 2)(x + 2)
Step 3: Identify the common factors
The common factor is (x - 2).
Step 4: Multiply together the common factors
LCD = (x - 2)(x + 2)
Step 5: Simplify the LCD
In this case, the LCD is already in its simplest form.
So, the least common denominator (LCD) for the rational expressions A and B is (x - 2)(x + 2).
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1.5r+15=2.25 just need help to show steps
Answer:
Step-by-step explanation:
Step 1: Subtract 15 from both sides.
1.5r+15−15=2.25−15
1.5r=−12.75
Step 2: Divide both sides by 1.5.
1.5r
1.5
=
−12.75
1.5
r=−8.5
Answer: r=-8.5
Step-by-step explanation:
1.5r+15=2.25
1.5r+15-15=2.25-15
1.5r=-12.75
1.5r/1.5=-12.75/1.5
r=-8.5
stock x has a beta of 0.7 and stock y has a beta of 1.3. the standard deviation of each stock's returns is 20%. the stocks' returns are independent of each other, i.e., the correlation coefficient, r, between them is zero. portfolio p consists of 50% x and 50% y. given this information, which of the following statements is correct?
Out of the given statements, statements number e) Portfolio P has the same required return as the market (rM) is the correct one
since we are given the information that stock x has a beta of 0.7 and stock y has a beta of 1.3 and the standard deviation of each stock's returns is 20%, and the correlation coefficient(r) between them is zero and at last, the portfolio p consists of 50% x and 50% y .So the beta of the portfolio P will be 0.5 * 0.7 + 0.5 * 1.3 = 1.0 .Now the return on the portfolio will be:
= risk-free rate + beta * (Expected market return - risk-free rate), Substituting the know values, we get,
= risk-free rate + 1 * (Expected market return - risk-free rate)
= Expected market return.
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Sock x has a beta of 0.7 and stock y has a beta of 1.3. the standard deviation of each stock's returns is 20%. the stocks' returns are independent of each other, i.e., the correlation coefficient, r, between them is zero. portfolio p consists of 50% x and 50% y. given this information, which of the following statements is correct?
a)Portfolio P has a standard deviation of 20%.b. The required return on Portfolio P is equal to the market risk premium (rM ? rRF).
c. Portfolio P has a beta of 0.7.
d. Portfolio P has a beta of 1.0 and a required return that is equal to the riskless rate, rRF.
e. Portfolio P has the same required return as the market (rM).
Pls help i will give brainiest
Answer:
red
Step-by-step explanation:
Answer:
red
Step-by-step explanation:
The y-intercept is -2 and the slope is 3/1. so you start on the y-axis at -2 and move up 3 to the right one.
An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard
The total number of rectangles you can draw using the grid lines exists 1296.
What are the peculiarities of rectangle?A quadrilateral is a shape with two opposed sides that are equal and parallel. It is a polygon with four sides and four 90-degree angles. The shape of a rectangle is two dimensional.
A square is a particular type of closed figure that has four straight sides, four right angles, and sides that are all the same length. As a result, we can say that: A square is a particular variety of rectangle. Rectangles are not always squares, but squares are always rectangles.
Number 2 × 2 squares have 4 black squares
all 3 × 3 squares have at least 4 black squares, as do all 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 squares.
there are 6 × 6 = 36 3 × 3 squares
there are 5 × 5 = 25 4 × 4 squares
there are 4 × 4 = 16 5 × 5 squares
there are 3 × 3 = 9 6 × 6 squares
there are 2 × 2 = 4 7 × 7 squares
there is 1 8 × 8 square
Summing we get 91 squares
The total number of rectangles you can draw using the grid lines exists (92)(92) = 1296.
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The total number of rectangles you can draw using the grid lines exists 1296.
What are the peculiarities of rectangle?A shape with two opposing sides that are equal and parallel is called a quadrilateral. It has four sides, four 90-degree angles, and is a polygon. A rectangle is a two-dimensional form.
An example of a closed form with four straight sides, four right angles, and all sides being the same length is a square. Therefore, we can state that a square is a certain type of rectangle. Squares are not always rectangles, but rectangles are always squares.
Number 2 × 2 squares have 4 black squares
all 3 × 3 squares have at least 4 black squares, as do all 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 squares.
there are 6 × 6 = 36 3 × 3 squares
there are 5 × 5 = 25 4 × 4 squares
there are 4 × 4 = 16 5 × 5 squares
there are 3 × 3 = 9 6 × 6 squares
there are 2 × 2 = 4 7 × 7 squares
there is 1 8 × 8 square
Summing we get 91 squares
The total number of rectangles you can draw using the grid lines exists (92)(92) = 1296.
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A first-time homebuyer is financing a house for $128,900. the buyer has to pay $300 plus 1.05% for a brokerage fee. how much are the mortgage brokerage fees? a. $1,653.45 b. $1,353.45 c. $1,356.60 d. $1,365.45
The brokerage charge he has to pay will be $1,659.45.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, A first-time homebuyer is financing a house for $128,900.
The brokerage charge = 300 + 1.05% of house charge
The brokerage charge = 300 + 1.05% of 128,900
The brokerage charge = 300 + 1.05/100 * 128,900
The brokerage charge = 300 + 1353.45
The brokerage charge = 1,653.45
therefore, the brokerage charge he has to pay will be $1,659.45.
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Find the missing value to the nearest hundredth.
Answer:
Option (A)
Step-by-step explanation:
Let the Sine of the given angle θ is,
Sinθ = \(\frac{x}{y}\)
Then θ = \(\text{Sin}^{-1}(\frac{x}{y} )\)
From the given values in the question,
Sinθ = \(\frac{7}{29}\)
θ = \(\text{Sin}^{-1}(\frac{7}{29} )\)
θ = 13.97°
Therefore, measure of the angle 'θ' will be 13.97°.
Option (A) will be the answer.
Which polynomial is factored completely? g Superscript 5 Baseline minus g 4 g cubed + 18 g squared + 20 g 24 g squared minus 6 g Superscript 4 2 g squared + 5 g + 4
Answer:
It's D on Edge 2021
I did the assignment
The polynomial that is factored completely is one that can not be factored again is 2g² + 5g + 4. Therefore, option D is the correct answer.
What is the factorisation of a polynomial?Factorisation of a polynomial is the decomposition of a polynomial into smaller polynomials.
Thus, the polynomial that is factored in completely is the polynomial that can not be decomposed again.
Now,
g⁵ - g = g(g⁴ - 1) (it can be factored)
4g³ + 18g² + 20g = 2g(g + 2)(2g + 5)
24g² - 6g⁴ = 6g²(4g² + g²)
2g² + 5g + 4 cannot be factored again. It is already factored completely
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Who can answer this??
Ths question is not complete but I will try my best to guide you through it so you can answer it. So what the question is asking you to do is that you have to plug in the coordinates into the equation. I don't want to give you an incorrect answer so I will explain it and you can do it yourself with the full question. First use the second equation as it is a full equation with an equal sign. Once plugged in you can find that c and d do not make the second equation true. A and B both complete the second equation. This means that either A or B is the answer. Since the first equation does not have an equal sign meaning it is not complete I can not answer it and I am sorry. But I have cut the answer down for you to A and B. If this answer helped you in the slightest please check out my channel The Game Rater. Thank you and I appreciate it.
-The Game Rater(Nitroid444)Can you explain why this is wrong? And what is the correct answer?
Area of a regular hexagon = perimeter x apothem x 1/2
perimeter = 6 x 18 in = 108 in.
Now to calculate the apothem, we have:
Now, using pythagoras theorem:
ap = √ (18² - 9²)
ap = 15.59 in
So, the area of the regular hexagon =
108 x 15.59 x 1/2 = 842 in²
Aline'sslopeis5,andits
y -intercept
is
3. What
isitsequationin
slope -intercept
form?
The equation of the line in slope-intercept form is: y = 5x + 3.
What is the Equation of a Line in Slope-intercept Form?If the value of the slope of a line, m, is known, and the value of its y-intercept, b, is known, the equation of the line in slope-intercept form would be written as:
y = mx + b.
Given the following:
Slope (m) of the line (m) = 5Y-intercept of the line (b) = 3To write the equation of the line in slope-intercept form, substitute m = 5 and b = 3:
y = 5x + 3
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Question:
A line's slope is 5, and its y -intercept is 3. What is its equation in slope-intercept form?
Given g of x equals cube root of the quantity x minus 3, on what interval is the function positive?
The function g(x) = ∛(x -3) is positive at the interval x > 3
Complete questionGiven g(x) = ∛(x -3), on what interval is the function positive?
How to determine the positive interval?The function is given as:
g(x) = ∛(x -3)
Set the radicand to positive
x - 3 > 0
Add 3 to both sides
x > 3
Hence, the function is positive at the interval x > 3
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Evaluate the integral ∫cos(t)/√(1 + sin^2(t)) dt
Show all steps
Therefore, the integral evaluates to sin(t) + C, where C is the constant of integration.
Dividing both sides by (1 - v²), we get:
u² = v² / (1 - v²)
Taking the square root, we have:
Since v = u/√(1 + u²), we substitute this expression into the equation to solve for u:
u = √((u/√(1 + u²))² / (1 - (u/√(1 + u²))²))
Simplifying, we have:
u = √(u² / (1 + u²) / (1 - u² / (1 + u²)))
= u / √(1 + u^2)
Since u = sin(t), we have:
sin(t) = u / √(1 + u²)
Multiplying both sides by √(1 + u²), we get:
√(1 + u²)sin(t) = u
Using the original substitution u = sin(t), we have:
√(1 + sin²(t))sin(t) = sin(t)
Therefore, v = sin(t).
Going back to the integral ∫(1/√(1 + u²)) du = v + C, we have:
∫cos(t)/√(1 + sin²(t)) dt = v + C = sin(t) + C
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! anybody know these answers ?
Answer:
i dont know sorry
Step-by-step explanation:
Answer:
1) $324
2) $630
3) $1254
4) $647.50
Step-by-step explanation:
1) $9.00hr times 36 hours = $324
2) $14.00 hr times 45 hours= $630
3) $22 hr times 57 hours = $1254
4) $18.50 hr times 35 hours =$647.50
The US consumes an average of 5.25 million metric tons of bananas per year. There are 317 million people in the US and there are 1000 kg in 1 metric ton. It costs $2.10 per kilogram of bananas. And how much money will this be per person every day?
Consumption of Bananas per year = 5, 250, 000 metric tons
We know 1000 kg = 1 metric ton, so
Consumption of Bananas per year (in kg) = 5, 250, 000 x 1000 = 5, 250, 000, 000 kgs
Cost of Banana = $2.10 per kg
So,
Cost of all bananas = 5, 250, 000, 000 x 2.10 = $ 11,025,000,000 (per year)
Taking a year to be 365 days, let's find the cost for a day:
\(\frac{11,025,000,000}{365}=\$30,205,479.4521\)Since there are 317 million [317,000,000] people in the US, the cost per person every day is:
\(\frac{30,205,479.4521}{317,000,000}\approx\$0.0952854\)Rounding to the nearest cent,
$0.10So, the amount (in dollars) per person every day = $0.10Answer$0.10every angle of a equilateral triangle is_____.
Please answer quick ill give brainliest to the best
Jerome parked his car for 4½ hours each day for two days at two different parking lots. The cost of parking at each parking lot is shown below. Day 1 - $5.00 for the first hour plus $1.50 for every additional hour. Day 2 - $2.50 for the first hour plus $1.50 for every additional hour. How much more money did Jerome pay on Day 1 compared to Day 2?
A) $19.00
B)$3.50
C) $18.00
D) $2.50
Tom bought 3 1/2 gallons of ice cream. If a pint is 1/8 of a gallon, how many pint-sized servings are there
Data were collected on the distance a golf ball will travel when hit by a golf club at a certain speed. the speed, s, is measured in miles per hour, and distance, y, is measured in yards. the regression line is given by Å· = 5.32 46.73s. identify the slope and y-intercept of the regression line. interpret each value in context. the slope, b = 5.32, indicates that the distance increases by 5.32 yards for every one mile per hour of speed. the y-intercept, a = 46.73, is the distance estimated by this model if the speed is zero miles per hour. the slope, b = 5.32, indicates that the distance decreases by 5.32 yards for every one mile per hour of speed. the y-intercept, a = 46.73, is the distance estimated by this model if the speed is one mile per hour. the slope, b = 46.73, indicates that the distance increases by 46.73 yards for every one mile per hour of speed. the y-intercept, a = 5.32, is the distance estimated by this model if the speed is zero miles per hour. the slope, b = 46.73, indicates that the distance decreases by 46.73 yards for every one mile per hour of speed. the y-intercept, a = 4.17, is the distance estimated by this model if the speed is one mile per hour.
The slope, b = 5.32, indicates that the distance increases by 5.32 yards for every one mile per hour of speed. The y-intercept, a = 46.73, is the distance estimated by this model if the speed is zero miles per hour.
Based on the given conditions, formulate,
The speed, s, is measured in miles per hour, and distance, y, is measured in yards.
The regression line is given by,
Å = 5.32 + 46.73s (Equation-1)
Then find,
The slope and y-intercept of the regression line.
The general form of linear regression line is
A linear regression line has an equation of the form Y = a + b x, where x is the explanatory variable and Y is the dependent variable. The slope of the line is b, and a is the intercept (the value of y when x = 0).
y = a + b x (Equation-2)
where x is variable. In the given equation the variable is denoted by s.
On comparing Equation-1 and Equation-2,
We can write,
a = 46.73
b = 5.32
It means slope is 5.32 and y-intercept is 46.73.
The slope, b = 5.32, indicates that the distance increases by 5.32 yards for every one mile per hour of speed.
The y-intercept, a = 46.73, is the distance estimated by this model if the speed is zero miles per hour.
Therefore,
The slope, b = 5.32, indicates that the distance increases by 5.32 yards for every one mile per hour of speed. The y-intercept, a = 46.73, is the distance estimated by this model if the speed is zero miles per hour.
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six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives. of the six countries, if country a sent the second greatest number of representatives, did country a send at least 10 representatives?
One of the six countries sent 41 representatives to the congress.
What, in the simplest words, is the Congress?
The Congress is our country's legislative body, which drafts our laws. The United States Senate and the United States House of Representatives are the two branches of Congress.
Each state has two U.S. Senators and a minimum of one U.S. Representative; the number of U.S. Representatives that can be elected depends on the number of citizens in the state.
(1) One of the six countries sent 41 representatives to the congress
(2) Country A sent fewer than 12 representatives to the congress
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Consider the function f(x) = x2 - 4x + 8 on the interval 0, 4. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval f(x) is f(x) is and f(0) on [0, 4] on (0, 4) f(4) Then by Rolle's theorem, there exists at least one value c such that f'(c) = 0. Find all such values c and enter them as a comma-separated list. Values of c =: (1 point) Consider the function graphed below. P n ? Does this function satisfy the hypotheses of the Mean Value Theorem on the interval a, b ? Does it satisfy the conclusion?? f(b) f(a)2 At what point c is f'(c) b - a
Verifying that this function satisfies the three hypotheses of Rolle's Theorem on the inverval f(x) is f(x) is and f(0) on [0, 4] on (0, 4) f(4) Then by Rolle's theorem, there exists at least one value c = 2 such that f'(c) = 0.
Given:
Consider the function f(x) = x2 - 4x + 8 on the interval 0, 4. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval f(x) is f(x) is and f(0) on [0, 4] on (0, 4) f(4) Then by Rolle's theorem, there exists at least one value c such that f'(c) = 0.
f(x)=x^2−4x+8, [0,4]
when, x = 0
f(x) = x^2 -4x +8
f(0) = y = 0 - 0 + 8 = 8
when, x=4
f(5) = y = 16 - 16+8 =
thus, we have 2 points (0, 8) ; (4, 8)
slope,m = {8-(8)} / {4-0} = 0
hence, we have to calculate all the points,x where 0<x<8 and slope=0
f '(x) = 2x - 4 = 0
or, f '(c) = 2c - 4 = 0
c = 4/2 =2 ( 0<x<4)
hence, the there is only one solution c=2 which satisfies Rolle's theorem.
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alculate the ????∘ for the following equation. use these standard potentials .fe(s) f2(g)⟶fe2 (aq) 2f−(aq)
The standard cell potential Ecell∘ for the equation Fe(s) + F₂(g) ⟶ Fe²⁺(aq) + 2F⁻(aq) is 2.87 V.
To calculate the standard cell potential (Ecell∘), we need to use the standard reduction potentials (E°) for the half-reactions involved in the cell. The reduction potential for the reduction half-reaction of Fe²⁺(aq) + 2e⁻ ⟶ Fe(s) is given as +0.44 V (taken from the standard reduction potentials table). The oxidation half-reaction of F₂(g) ⟶ 2F⁻(aq) + 2e⁻ has a reduction potential of +2.87 V.
To find the overall cell potential, we subtract the reduction potential of the anode (F₂) from the reduction potential of the cathode (Fe²⁺/Fe):
Ecell∘ = Ered(cathode) - Ered(anode)
Ecell∘ = (+0.44 V) - (+2.87 V)
Ecell∘ = -2.43 V
Since the standard cell potential is negative (-2.43 V), it indicates that the reaction is not spontaneous under standard conditions.
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the complete question is:
Calculate the standard cell potential Ecell∘, for the equation
Fe(s)+F2(g)⟶ Fe2+(aq)+2F−(aq)
Standard reduction potentials can be found in this table.
Ecell∘=V
Please help asap, my semester ends in less then 2 weeks and I’m struggling
The probability that, in a random sample of 6 parts produced by this machine, exactly 1 is defective is 0.371.
How to calculate the probabilityIn this case, we have n = 6 (the number of parts) and p = 0.13 (the probability of producing a defective part). We want to find the probability of exactly 1 defective part, so k = 1.
Plugging in the values into the formula, we get:
P(X = 1) = C(6, 1) * 0.13 * (1 - 0.13)⁵
= 6 * 0.13 * 0.87⁵
Calculating this expression:
P(X = 1) ≈ 0.371
Therefore, the probability that, in a random sample of 6 parts produced by this machine, exactly 1 is defective is approximately 0.371
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At a certain auto parts manufacturer, the Quality Control division has determined that one of the machines produces defective parts 13% of the time. If this percentage is correct, what is the probability that, in a random sample of 6 parts produced by this machine, exactly 1 is defective?
Round your answer to three decimal places.
If a 50 kg object is at a location 25,600 km from Earth’s center, what is the gravitational force exerted by the object on Earth? In what direction does that force act?
Answer:
Approximately \(30\; \rm N\) toward the center of the object.
Step-by-step explanation:
Look up the gravitational constant and the mass of the earth:
\(\begin{aligned}G &\approx 6.67 \times 10^{-11}\; \rm m^{3} \cdot kg^{-1} \cdot s^{-2} \\ &= 6.67 \times 10^{-11} \; \rm N \cdot m^{2} \cdot kg^{-2}\end{aligned}\).
\(M \approx 5.97 \times 10^{24}\; \rm kg\).
Convert the unit of distance to standard units:
\(\begin{aligned}r &= 25600\; \rm km \\ &= (25600 \times 10^{3})\; \rm m \\ &= 2.56 \times 10^{7}\; \rm m\end{aligned}\).
Consider two spherical objects with mass \(m_{1}\) and \(m_{2}\). Let \(r\) denote the distance between the centers of the two objects. By Newton's Law of Universal Gravitation, the size of the gravitational force between these two objects would be:
\(\begin{aligned}F &= \frac{G \cdot m_{1} \cdot m_{2}}{r^{2}}\end{aligned}\).
Thus, the size of the gravitational force between this object and the earth would be:
\(\begin{aligned}F &= \frac{G \cdot m \cdot M}{r^{2}} \\ &\approx \frac{1}{2.56\times 10^{7}\; \rm m} \\ &\quad \times 6.67 \times 10^{-11}\; \rm N \cdot m^{2}\cdot kg^{-2} \\ &\quad \times 50\; \rm kg \times 5.97 \times 10^{24}\; \rm kg \\ &\approx 30\; \rm N\end{aligned}\).
Since gravitational force is attractive, the \(50\; \rm kg\) object in this question would attract the earth through the gravitational force. Thus, this force would point towards the center of this \(50\; \rm kg\!\) object.
x - 3 < 4
graph the inequality
Answer: See the image below.
The y-intercept in the linear equation y=−1/10x−2 is _[blank]_.
Enter your answer as the value that correctly fills in the blank, like this: 42
If your answer is a fraction, enter it formatted like this: 42/53
Answer:
-2 is the answer
Step-by-step explanation:
y=mx+b
b= y intercept
because it is a negative it replaces the addition sign so that it is not a positive 2.
Line L is tangent to the graph of y = ex at the point (k, ek y-intercept of L is 1/2 (A) 0.405 (B) 0.768 (C) 1.500 (D) 1.560 (E) There is no such value of k
Since x > 0, the tangent of f'(x) is always positive, and the denominator is always positive. Therefore, f'(x) is always positive, which means that f(x) is always increasing and can only cross the x-axis once. Hence, f(x) has only one root on its entire domain.
To find the y-intercept of line L, we need to determine the equation of line L.
Since line L is tangent to the graph of y = ex at the point (k, ek), the slope of line L must be equal to the slope of the tangent line to y = ex at (k, ek), which is simply ek.
Therefore, the equation of line L is:
y - ek = ek(x - k)
Simplifying, we get:
y = ekx - ek2
To find the y-intercept, we set x = 0 and solve for y:
y = ek(0) - ek2 = -ek2
Therefore, the y-intercept of line L is -ek2.
We need to find the value of k such that the y-intercept of line L is 1/2.
-ek2 = 1/2
Solving for k, we get:
k = ln(sqrt(2))
So the answer is (A) 0.405.
To prove that the function has only one root on its entire domain, we take the derivative of f(x) and show that it is always positive.
f(x) = ln(x) - 1/x
f'(x) = 1/x + 1/x^2 = (x+1)/x^2
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for rotation, when swapping the x and y and making the y negative, what happens if the new y is already negative
When you perform a 2D rotation of a point by swapping the x and y coordinates and negating the new y coordinate, the sign of the y-coordinate will change based on the quadrant in which the original point lies.
If the original point lies in the first or fourth quadrant, the new y-coordinate will be negative after rotation. In this case, if the new y-coordinate is already negative, then you do not need to negate it again.
On the other hand, if the original point lies in the second or third quadrant, the new y-coordinate will be positive after rotation. In this case, if the new y-coordinate is already negative, then you need to negate it to correctly represent the rotation.
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