The simplification of the expression m²n³+ 18m³n² + 10m²n+2m²n is m²n ( n²+18mn+ 12)
What is simplifying expression?Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do.
m²n³+ 18m³n² + 10m²n+2m²n = m²n³ + 18m³n² +12m²n
Therefore we factorize the common factors out, they are m²n
= m²n ( n²+18mn+ 12)
therefore the simplification of the expression m²n³+ 18m³n² + 10m²n+2m²n is m²n ( n²+18mn+ 12)
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On another day, Martin bought 12 3/5
pounds of grapes for a picnic. His friend bought 3/
8
of that amount. Use compatible fractions to estimate how many pounds of grapes Martin’s friend bought.
The friend bought 4.725 pounds of grape, which is 3/8 of the amount that Martin bought
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on
Mixed fraction is a fraction that is made up of a whole number and a fraction combined together.
Martin bought 12 3/5 pounds of grapes for a picnic.
12 3/5 pound is a mixed fraction
12 3/5 pound = 63/5 pounds
His friend bought 3/8 of that amount. Hence:
Amount of grapes the friend bought = (3/8) * 63/5 pounds = 4.725 pounds
The friend bought 4.725 pounds of grape
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Sofia has oranges and watermelons in a ratio of 700:500. How many oranges does she have if she has 5 watermelons?
Sofia has 7 oranges
What is ratio?Ratio is a term that is used to compare two or more numbers. It is used to indicate how big or small a quantity is when compared to another.
The ratio of 700:500 can be written as 700/500
The ratio can be reduced to the lowest terms for easy calculation
We have 7/5
Let x represent orange
7/5 = x/5
By cross multiplying
35 =: 5x
Divide both sides of the equation by 5
35/5 = 5x/5
X = 7
Hence, if she has 5 watermelons, she will has 7 oranges
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find the sign of the expression if the terminal point determined by t is in the given quadrant.
The expression tan(t)sin(t)/cot(t) is negative.
What is the quadrant?
A quadrant is one of the four regions into which the plane is divided by the x-axis and y-axis. There are four quadrants in total:
First quadrant (Quadrant I): The values of x and y are both positive.
Second quadrant (Quadrant II): The value of x is negative, and the value of y is positive.
Third quadrant (Quadrant III): Both the values of x and y are negative.
Fourth quadrant (Quadrant IV): The value of x is positive, and the value of y is negative.
The location of a point in a plane can be determined by its coordinates (x, y), and based on the sign of x and y, the point can be located in one of the four quadrants.
The given expression is tan(t)sin(t)/cot(t).
If the terminal point determined by t is in quadrant III (180° < t < 270°), then tan(t) and sin(t) are negative, and cot(t) is positive. In this case, the expression tan(t)sin(t)/cot(t) is negative.
Hence, the expression tan(t)sin(t)/cot(t) is negative.
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5. Mr. Muscles loads up a bar with 910 newtons (~205 lbs) of weight and pushes the
bar up over his head eight times. Each time he lifts the weight .5 meters. How much
work did he do If he does the whole thing in 15 seconds, how much power did it
take? SHOW YOUR WORK.
It took 245.33 watts of power to perform the work.
What is the work?
Work is defined as a force causing the movement or displacement of an object. In the case of a constant force, work is the scalar product of the force acting on an object and the displacement caused by that force.
The work done by Mr. Muscles can be calculated as follows:
Work = Force x Distance x Cos(θ)
where Force is 910 N, Distance is 4 m (since he lifts the weight 0.5 m 8 times), and θ is the angle between the force applied and the displacement of the weight, which is assumed to be 0° in this case since the weight is being lifted straight up.
Work = 910 N * 4 m * cos(0°) = 3640 N * m
The power required to perform this work in 15 seconds can be calculated as follows:
Power = Work / Time
Power = 3640 N * m / 15 s = 245.33 W
hence, it took 245.33 watts of power to perform the work.
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Which equation is equivalent to the formula below?
The equivalent equation will be a = (y - k) / (x - h)². Then the correct option is C.
What is an equivalent?The equivalent is the expression that is in different forms but is equal to the same value.
The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.
The equation is given below.
y = a(x - h)² + k
Simplify the equation for 'x', then we have
y = a(x - h)² + k
y - k = a(x - h)²
a = (y - k) / (x - h)²
The equivalent equation will be a = (y - k) / (x - h)². Then the correct option is C.
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suppose a firm is initially producing 250 units of output at point c, where the green line labeled
The firm is initially producing 250 units of output at point c, where the green line labeled "marginal cost" intersects with the red line labeled "average total cost."
This means that the marginal cost of producing 250 units of output is equal to the average total cost of producing 250 units of output.
Mathematically, the marginal cost of producing 250 units of output is equal to the change in total cost divided by the change in quantity, or MC = (TC₂ - TC₁)/(Q₂ - Q₁). In this example, the marginal cost of producing 250 units of output is equal to the average total cost, which is TC₂/Q₂. Therefore, the marginal cost of producing 250 units of output is equal to the average total cost at point c.
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help me please!!!!!!!!!!!!!!
quality ratings of airports: the international air transport association surveys business travelers to develop quality ratings for transatlantic gateway airports. the maximum possible rating is 10. suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the miami international airport. the file miami contains the ratings obtained from the sample of 50 business travelers. develop a 95% confidence interval estimate of the population mean rating for miami.
To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:
Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).
Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.
Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).
Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.
Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.
The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.
It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.
Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.
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To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:
Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).
Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.
Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).
Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.
Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.
The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.
It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.
Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.
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How do you write the ratio 12:3 in simplest form
Answer: 4:1, 4/1 or 4 to 1,
Step-by-step explanation:
If we divide 12 by 3, we would get the answer 4 as the numerator. And then we would divide 3 by 3, which would give us 1 as the denominator.
Suppose you paid $2.82 sales tax for a sweatshirt in state where the
sales tax is 6%. What was the original price of the sweatshirt?
Answer:
$47
Step-by-step explanation:
First, we have to find 1% by taking
2.82 divided by 6 = $0.47
To find the original, we take
0.47 times 100 = $47
So, the original price of the sweatshirt is $47
The original price of the sweatshirt is, $47
What is mean by Percentage?A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
To Calculate the percent of a number , divide the number by whole number and multiply by 100.
Given that;
Suppose you paid $2.82 sales tax for a sweatshirt in state where the sales tax is 6%.
Let the original price of the sweatshirt is, x
Hence, We can formulate;
6% of x = 2.82
6x / 100 = 2.82
6x = 2.82 x 100
6x = 282
x = 282 / 6
x = $47
Thus, The original price of the sweatshirt is, $47
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(a) Write an integral which represents the area between f(x)= x^4 and the x-axis, between x = 0 and x = 2.
(b) Evaluate this integral using the Fundamental Theorem of Calculus (the Evaluation Theorem).
The integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.
What is Fundamental Theorem of Calculus?A theorem that connects the ideas of integrating and differentiating functions is known as the fundamental theorem of calculus. By calculating the difference between the antiderivative at the higher and lower limits of the integration process, the fundamental theorem of calculus supports the method.
The given function of the area is:
f(x) = x^4.
The integral which represents this area can be written as:
[tex]A = \int\limits^2_0 {x^4} \, dx[/tex]
Using the fundamental Theorem of Calculus we have:
[tex]A = \int\limits^2_0 {x^4} \, dx\\\\A = [\frac{x^5}{5} ]_0^2[/tex]
Substituting the value of limits we have:
A = [2^5 / 5] = 32/5
Hence, the integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.
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find the general solution of the following differential equation. primes denote derivatives with respect to x.
3X^2y'+6xy=15y^3
The General solution is ...
The general solution to the differential equation [tex]3x^2y' + 6xy = 15y^3[/tex] is not possible to obtain in a closed form as it is a nonlinear differential equation.
One way to find a numerical solution to this equation is to use numerical methods such as the Runge-Kutta method, the Euler method, or finite difference methods. Another way is to use numerical software, such as MATLAB, to obtain an approximate solution. To find an analytical solution, one can try to transform the equation into a more manageable form using substitution or other techniques, but it may not be possible to obtain an exact solution. The general solution to the differential equation [tex]3x^2y' + 6xy = 15y^3[/tex] is not possible to obtain in a closed form as it is a nonlinear differential equation.
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suppose that two shuttle buses operate (without coordination) between the campus and downtown. one bus can carry 20 people and has a round trip time of 8 min (including loading and unloading passengers), the other can carry 25 people but has a round trip time of 10 min. there is always a queue of people waiting at the downtown bus stop, but passengers come to the campus bus stop at a steady rate of 2/min. if neither bus is idle at any time, determine:
The complete answers regarding flow of people and occupancy are solved as follows:
(a) The flow of people from downtown to campus is the rate at which the shuttle buses can transport passengers from the downtown bus stop to the campus bus stop.
(b) The flow of people from campus to downtown is equal to the rate at which passengers are coming to the campus bus stop.
(c) The average bus occupancy from campus to downtown is equal to the number of passengers on each bus divided by the number of seats on each bus.
Dispatching the buses from downtown to campus at equal headways would reduce the average waiting time of passengers going from campus to downtown. This is because the slower bus would have more time to pick up and drop off passengers, and the faster bus would not be slowed down by having to wait for the slower bus. The overall result would be a reduction in the average waiting time for all passengers.
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Complete Question:
Suppose that two shuttle buses operate (without coordination) between the campus and downtown. One bus can carry 20 people and has a round trip time of 8 min (including loading and unloading passengers), the other can carry 25 people but has a round trip time of 10 min. There is always a queue of people waiting at the downtown bus stop, but passengers come to the campus bus stop at a steady rate of 2/min. If neither bus is idle at any time, determine:
(a) The flow of people from downtown to campus.
(b) The flow of people from campus to downtown.
(c) The average bus occupancy from campus to downtown.
If one insisted on dispatching these buses from downtown to campus at equal headways, we would only slow down the faster bus. Would you expect this strategy to reduce the average waiting time of passengers going from campus to downtown?
What is the distance between points e and g? round to the nearest tenth of a unit. Responses 11. 2 units 11. 2 units 12. 0 units 12. 0 units 14. 3 units 14. 3 units 17. 0 units.
The angle of separation between E and G is 12.0415945788 degrees.
How do coordinate points work?A set of numbers that expresses "the position of points on a coordinate plane utilising the horizontal and vertical distances from two reference axes."Both Point E's and Point G's coordinates are the first pair in an ordered sequence (1,7). (9,-2).A point or shape's location in a two-dimensional plane can be found using coordinates, which are a pair of numbers. The terms "x-coordinate" and "y-coordinate" are used to define a point's location on a 2D plane. The origin of the coordinate system is the intersection of the axes, and the first and second coordinates are known as the abscissa and the ordinate of P, respectively. The coordinates are typically expressed as two numbers enclosed in parentheses and placed in that order, separated by a comma, as in (3, 10.5).Distance formula =[tex]$\sqrt{\left(\text { dif ferenceof } f^{\prime} x^{\prime} \text { points }\right)^2+\left(\text { Differenceof }{ }^{\prime} y^{\prime} \text { points }\right)^2} \\[/tex]
[tex]$& =\sqrt{(1-9)^2+(7-(-2))^2} \\[/tex]
[tex]& =\sqrt{(-8)^2+(9)^2} \\[/tex]
[tex]& =\sqrt{64+81} \\[/tex]
[tex]& =\sqrt{145} \\[/tex]
= 12.0415945788
As a result, the distance between E and G is 12.0415945788 miles.
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what is the difference between constructing a regular hexagon and constructing an equilateral triangle? (1 point)
Answer: Your answer is A or
For the hexagon construction, every arc along the circle is connected, but for the equilateral triangle construction, every other arc along the circle is connected.
Step-by-step explanation: It makes sense because in the lesson you need to make 6 arcs for both hexagons and equilateral triangles. However hexagons have 2 times as much sides as triangles so to make a triangle you would have to make a line with every other arc. That means skip an arc each line you make. For A hexagon obviously because it has 6 sides you just draw the lines to connect all 6 arcs
Hope this helps!
What is the Meaning of Arc?
An arc in mathematics is a segment of a curve or a segment of a circle.
What is an arc?
A circle's arc is referred to as a portion or section of its circumference. A semicircular arc is one whose length is exactly half that of a circle.
The arc is represented by the letters ‘⌒’ or ‘⌢’ in Euclidean geometry.
There are two methods for measuring an arc which are:
1. Length of the arc
The length of the arc is calculated in units of distance, such as centimeters. The arc is preceded by the lowercase letter l (for "length") to identify it.
2. Angle of the arc
The angle of the arc is the angle that the arc at the centre of the circle subtends.
Hence, an arc in mathematics is a segment of a curve or a segment of a circle.
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If the yield to maturity on an annual-pay bond is 7.75%, the bond-equivalent yield isclosestto:A. 7.61%.B. 7.90%.C. 8.05%.
The bond yield that comes closest to 7.75%, the amount yield to maturity on an annual-pay bond, is 7.61%.
A. The bond yield that comes closest to 7.75%, the yield to maturity on an annual-pay bond, is 7.61%. A measure of return that considers the market parameters used to determine the return from an investment in a bond is called the bond-equivalent yield. It accounts for the time value of money, the coupon rate, and the current market rate to determine the yearly rate of return that would be generated if the bond were kept for a year. When comparing bonds with differing coupon rates and maturities, the bond-equivalent yield is utilised. It is a helpful tool for investors who want to compare various bonds and decide on their investments with knowledge.
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prove the following equalities using mathematical induction: a. ∑ ― 1 = 1 ( 1 i(i 1)) = 1 ― 1/, ≥ 2.
We can prove the equality using mathematical induction. We assume it holds for n and show it holds for n+1. Then, we add n+1 to both sides of the equation, proving the equality holds for n+1.
To prove the equality using mathematical induction, we must first assume it holds for n. Then, we can add n+1 to both sides of the equation:
∑ ― 1 = 1 ( 1 i(i 1)) + (n + 1) = 1 ― 1/ + (n + 1)
The right side of the equation is equal to 1 ― 1/ + n + 1, which simplifies to 1 ― 1/(n+1), proving the equality holds for n+1. Therefore, the equality holds for all n ≥ 2, and we have proven the equation using mathematical induction. Note: The expression in the equation should be n(n+1)/2, not 1/i(i+1).
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What is the derivative of an absolute value function?
The derivative of an absolute value function is the slope of the tangent line to the curve at a specific point.
What is an absolute value function?
A function in algebra with the variable inside the absolute value bars is called an absolute value function. The most popular form of the absolute value function is f(x) = |x|, where x is a real integer, and this function is sometimes referred to as the modulus function.
The slope of the tangent line to the curve at a specific point is the derivative of an absolute value function, or for that matter, of any function. Such functions can be differentiated individually since they are piecewise functions.
Hence, the slope of the tangent line to the curve at a specific point is the derivative of an absolute value function.
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You are building a ramp that must cover a horizontal distance of exactly 15 feet. The angle of the ramp from the ground is 12°. Determine the length of the ramp, in feet. Round to two decimal places as needed.
The length of the ramp is solved to be 15.33 feet
How to find the length of the rampThe length of the ramp is worked using SOH CAH TOA
Sin = opposite / hypotenuse - SOH
Cos = adjacent / hypotenuse - CAH
Tan = opposite / adjacent - TOA
The direction of movements describes a right triangle of
adjacent = 15 feet
hypotenuse = length of the ramp = ?
The length of the ramp is calculated using cos, CAH
let the angle be x
cos x = adjacent / hypotenuse
cos 12 = 15 / hypotenuse
hypotenuse = 15 / cos 12
hypotenuse = 15.334
length of ramp = 15.33 feet
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let f be the function defined by f(x) = (x2 1)e-x for -4 less than or equal to x less than or equalto four.a) for what values of x does f reach its absolute maximum?
Consequently, the interval's leftmost extreme is f(-4) = 17e^{-4}, where the absolute maximum is located.
The function f is f(x) = [tex](x^2 + 1)e^{-x}[/tex]
First, solve the equation f'(x) = 0 to identify the critical locations. It will be calculated using the formula [tex]\frac{d}{dx}(xy) = x\frac{d}{dx}(y) + y\frac{d}{dx}(x)[/tex].
f'(x) = [tex](x^2 +1)\frac{d}{dx}e^{-x}+e^{-x} \frac{d}{dx}(x^2+1)[/tex]
f'(x) = [tex]-(x^2 + 1)e^{-x}+2xe^{-x}[/tex]
Simplifying
Taking e^{-x} common
f'(x) = [tex][2x-(x^2+1)]e^{-x}[/tex]
Solving the bracket
f'(x) = [tex]e^{-x}(-x^2+2x-1)[/tex]
Now equating f'(x) = 0, then
[tex]e^{-x}(-x^2+2x-1)[/tex] = 0
The only essential point is at x = 1 since e^{-x} is zero only at x = ∞.
f(1) = 2e^{-1}.
Then determine the value that defines the interval's extremes:
f(-4) = 17e^{-4}
The maximal candidate is f(-4) since it is the largest of all the candidates.
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I NEED HELP!
The question asks to prove that this diagram is congruent. But when I do the steps, I can't find the correct answer.
I need step by step answer of this question. Thank you
Step-by-step explanation:
For two triangles to be congruent, they either need to have three respectively equal sides (SSS), two respectively equal signs with the same angle between them (SAS), two same angles with a respectively equal side between them (ASA), or for a right-angled triangel, a respectively equal hypotenuse and side (RHS).
Here we have the triangles ΔABC and ΔCDE. We know DC is equal to BC because they are marked as such, same for AC and EC. We also know <ACB = <DCE because they are vertically opposite angles. Since we have two triangles with equal sides and the same angle between these sides (SAS), we can say these triangles are congruent.
he line representing the equation part of the constraint 7x1 + 4x2 s 28 goes through the following two points (4,0) and (7,0) (0, 4) and (7,0) (4,0) and (0,7) (0, 4) and (0,7) none of the above
The line representing the equation part of the constraint [tex]$7 x_1+4 x_2 \leqslant 28$[/tex] goes through (4,0) & (0,7).
Hence, option (c) is the correct choice.
The constraint equation is g(x,y)=c, and we state that x and y are constrained by g(x,y)=c.
Constrained maximum and constrained minimum points are locations (x,y) that are maxima or minima of f(x,y) with the condition that they meet the constraint equation g(x,y)=c.
A constraint function can be translated into a new form that is equal to the original function; that is, the constraint boundary and feasible set for the problem remain unchanged, but the function's form does.
The given constraint is:
[tex]7 x_1+4 x_2 \leq 2[/tex]
If x_1=0
So, we will get:
4x_2=28
=>x=7
If x_2=0
7 x_1 =28
x_1 =4
So, the points are: (4,0)
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The sides of a triangle are 87, 59, and 79. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.
The Pythagorean Theorem shows that the triangle is acute
How to determine the true statementFrom the question, we have the following parameters that can be used in our computation:
Using the Pythagorean Theorem, we have
87^2 < 59^2 + 79^2
Evaluate
7569 < 9722
Since 7569 is less than 9722, we can conclude that the triangle is an acute triangle, not a right or obtuse triangle.
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42 points are placed inside a square with sides of length 2. how many points are guaranteed to be within of each other?
It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.
Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.
The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.
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It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.
Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.
The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.
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what is the common base for 5 to the power of 9 divided by 5 to the power of 6
a box has 4 green balls and 5 red balls. a number, i, is picked at random from {2,3,4,5}. then i balls are chosen at random from the box without replacement.
a) The probability that the 3 balls chosen were all green is 1/21.
b) The probability that all the balls are chosen is red is 1/252.
What is probability?
Probability is a mathematical concept that measures the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
(a) Given 3 balls were chosen at random without replacement, the probability that they were all green is given by:
P(green) = (number of favorable outcomes) / (number of total outcomes)
The number of favorable outcomes is the number of ways to choose 3 green balls out of 4, which is C(4,3) = 4.
The number of total outcomes is the number of ways to choose 3 balls out of 9, which is C(9,3) = 84.
So, the probability that 3 balls chosen were all green is:
P(green) = C(4,3) / C(9,3) = 4 / 84 = 1/21
(b) The probability that all the balls chosen are red is given by:
P(red) = (number of favorable outcomes) / (number of total outcomes)
The number of favorable outcomes is the number of ways to choose 5 red balls out of 5, which is C(5,5) = 1.
The number of total outcomes is the number of ways to choose 5 balls out of 9, which is C(9,5) = 126.
So, the probability that all the balls chosen are red is:
P(red) = C(5,5) / C(9,5) = 1 / 126 = 1/252
Hence,
a) The probability that the 3 balls chosen were all green is 1/21.
b) The probability that all the balls are chosen is red is 1/252.
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There are two noncongruent triangles where B = 55 degrees, a 15, b = 13 Find the measures of the angles of the triangle with the greater perimeter. Round to the nearest tenth if necessary
The triangle with the greater perimeter has angles 55, 15, and 112 degrees.
In a triangle, the sum of the interior angles is 180 degrees. Thus, if we have two non-congruent triangles with one angle equal (in this case, B = 55 degrees), we can set up an equation to solve for the third angle of the triangle with the greater perimeter.
Let's call the third angle in the first triangle x. Then, x + 55 + b = 180 degrees, or x + b = 125 degrees.
Since we know that a + b + B = 180 degrees for any triangle, we can set up a similar equation for the second triangle: y + 55 + a = 180 degrees, or y + a = 125 degrees.
Since the perimeter of a triangle is equal to the sum of its side lengths, and we know that a = 15 and b = 13, we can set up another equation to find the triangle with the greater perimeter:
Perimeter 1 = 15 + 13 + x and Perimeter 2 = 15 + 13 + y.
Solving for x, we find that x = 125 - b = 125 - 13 = 112 degrees. Solving for y, we find that y = 125 - a = 125 - 15 = 110 degrees.
Since 112 > 110, the triangle with the greater perimeter has angles 55, 15, and 112 degrees.
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The triangle with the greater perimeter is the one with the greater value of c, which we can calculate using the law of cosines and the law of sines.
Given two noncongruent triangles with angle B equal to 55 degrees and sides a and b, we need to find the measures of the angles of the triangle with the greater perimeter.
First, let's use the law of cosines to find the measure of angle A in each triangle:
[tex]c^2 = a^2 + b^2 - 2ab * cos(A)\\\\c = \sqrt{(a^2 + b^2 - 2ab * cos(A))} \\\\cos(A) = (a^2 + b^2 - c^2) / 2ab[/tex]
For the first triangle, we can use the given values:
[tex]cos(A) = (15^2 + 13^2 - c^2) / 2 * 15 * 13\\\\c = \sqrt{(15^2 + 13^2 - 2 * 15 * 13 * cos(A))}[/tex]
For the second triangle, we can use a and b as the lengths of the sides opposite angles A and C, respectively. Then we can use the law of sines to find the lengths of the other sides:
[tex]sin(A) / a = sin(C) / c\\\\c = a * sin(C) / sin(A)[/tex]
Substituting values and solving, we get:
[tex]c = 15 * sin(C) / sin(A)\\\\cos(A) = (15^2 + b^2 - c^2) / 2 * 15 * b\\\\c = \sqrt{(15^2 + b^2 - 2 * 15 * b * cos(A))}[/tex]
Now we can compare the two perimeters (the sum of the lengths of all sides) to determine which triangle has the greater perimeter. The perimeter of each triangle will be a + b + c.
Whichever triangle has the larger value of c will have the greater perimeter, as a and b are the same in both triangles.
So, we can use the value of c that we just calculated for each triangle to determine which triangle has the greater perimeter, and therefore, which triangle has the greater measure of its angles.
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What is the solution of the system of equations shown in the following graph?
find an implicit solution to the initial value problem. dy dx = x5 6 y2 y 1
[tex]y = 4x^5 + Cx-6[/tex] the implicit solution to the initial value problem is y = [tex]4x^5 + Cx-6[/tex], where C is an arbitrary constant.
Start by isolating the y-derivative on the left side of the equation.
[tex]dy/dx = x^5 6 y^2[/tex]
Divide both sides by the coefficient of the y-derivative (6y2).
[tex]1/6(dy/dx) = x5/6 y2[/tex]
Integrate both sides with respect to y.
[tex]1/6∫(dy/dx)dy = ∫(x5/6 y2)dy 1/6y + C1 = (x5/6)y3/3 + C2[/tex]
Rearrange the equation to solve for y.
[tex]y = 4x5 + Cx-6[/tex]
Finally, the implicit solution to the initial value problem is [tex]y = 4x5 + Cx-6,[/tex]where C is an arbitrary constant.
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