Answer: We can start by expanding the square on the left-hand side of the equation:
(s + t)^2 = s^2 + 2st + t^2
We know that s^2 + t^2 = 6, so we can substitute that into the equation:
(s + t)^2 = 6 + 2st
We also know that st = c + 5, so we can substitute that into the equation:
(s + t)^2 = 6 + 2(c + 5)
Simplifying the right-hand side:
(s + t)^2 = 6 + 2c + 10
So, in terms of c, (s + t)^2 = 6 + 2c + 10.
Step-by-step explanation:
According to AAA ('Triple' A), the median amount that Americans spent over the 2012 Labor Day Holiday was $761.00. Consider the pie chart below.
A -- Fuel Transportation
B -- Other Transportation
C -- Food and Beverage
D -- Shopping
E -- Entertainment/Recreation
F -- Hotel
G -- Other
Express your answers rounded to the nearest cent.
a). How much money did the average American spend on Food & Beverage over the 2012 Labor Day Holiday?
b). How much money did the average American spend on Entertainment/Recreation over the 2012 Labor Day Holiday?
c). How much more money did the average American spend on Food & Beverage than on Entertainment/Recreation over the 2012 Labor Day Holiday?
a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.
What is pie chart?In order to show mathematical issues, the "pie chart," often referred to as a "circle chart," divides the circular statistical visual into sectors or portions. A proportionate amount of the entire is indicated by each sector. The Pie-chart is now the most effective method for determining the composition of something. Pie charts typically take the role of other graphs, such as bar graphs, line plots, histograms, etc.
Given that, the average American spent $761.00.
a. Money spent on Food and Beverages is:
F = 761(20/100) = $152.2
b. Money spent on Entertainment is:
E = 761(13/100) = $98.93
c. The money spent on food is more than entertainment by:
A = 152.2 - 98.93 = $53.27
Hence, a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.
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The measure of one angle of a right triangle is 26°. Find the measure of the other angle.
Enter an integer or decimal number [more...]
Question Help:
Post to forum
Calculator
Answer:
64°
Step-by-step explanation:
the sum of the 3 angles in a triangle = 180°
let the third angle be x , then
x + 90° + 26° = 180°
x + 116° = 180° ( subtract 116° from both sides )
x = 64°
the other angle is 64°
Choose the property for each
Answer:
w = 12
Step-by-step explanation:
Simplifying equation:To simplify the equation, we have to isolate 'w'. To isolate 'w',
Add 21 to both sides of the equation. Multiply both sides by 2.[tex]\dfrac{w}{2}-21=-15\\\\\\\dfrac{w}{2 }-21+21=-15+21 \ \text{\bf (Addition property of equality)}\\[/tex]
[tex]\dfrac{w}{2}= 6[/tex]
[tex]2*\dfrac{w}{2}=6*2 \ \text{\bf (Multiplication property of equality)}\\\\\\[/tex]
w = 12
Each class you create becomes a new ______ that can be used to declare variables and create objects. a. package b. instance c. library d. type. ANS: d. type.
The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.
The formula for calculating a rate of return is expressed as the net investment income divided by the initial investment (ROI = (Gain from Investment – Cost of Investment) / Cost of Investment). To calculate the rate of return, you need to first determine the net investment income by subtracting the cost of investment from the gain from investment. Then, divide this number by the cost of investment and multiply it by 100 to convert it into a percentage. For example, if you invest $1000 and you make a total of 1200 back, your rate of return is 20%. This is calculated by subtracting the cost of investment (1000) from the gain (1200), resulting in a net income of 200. Then, divide $200 by the cost of investment (1000) to get 0.2 and multiply it by 100 to get a rate of return of 20%.
The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.
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HELP PLEASE! Find the surface area of the composite shape. Round to the nearest tenth.
The surface area of the composite figure is 415. 25 in²
How to determine the surface area of the composite structureFrom the image shown, we can see that the figure is made up of a cuboid and a semicircle.
The formula for calculating the surface area of a cuboid is given as;
Surface area =2lw+2lh+2hw
Given that, l is the length, h is the height and w, the width.
Surface area = 2(6×10) + 2(6×8) + 2(8 × 10)
expand the brakect
Surface area = 120 + 96+ 160
Surface area = 376 in²
Surface area of a semicircle = 1/2(πr2 ),
Substitute the values
Surface area = 1/2 ×22/7 ×5² = 39.25 in²
Hence, the value is the sum of the surface areas
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Mark had 5/6 as many birds stickers as bear stickers at first. After buying another 20 stickers of each type, he has 7/8 as many bird stickers as bear stickers. How many of each type of stickers did mark have at first?
Mark had 480 bear stickers and 400 bird stickers at first.
What is word problem?A word problem in math is a math question written as one sentence or more that requires children to apply their math knowledge to a 'real-life' scenario.
Given that, Mark had 5/6 as many birds stickers as bear stickers at first. After buying another 20 stickers of each type, he has 7/8 as many bird stickers as bear stickers.
We are asked to find the number of each type of stickers did mark have at first.
Let Mark have x bear stickers at first,
Therefore,
Bird sticker = 5x/6
5x/6 + 20 = 7x/8
Multiply the equation by 48
40x + 960 = 42x
2x = 960
x = 480
5x/6 = 400
Hence, Mark had 480 bear stickers and 400 bird stickers at first.
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point B is 2 root 5 units from A and is on the intersection of two gridlines write the co-ordniates of both positions of point B
The coordinates of both positions of point B is (2√5,0) and B2: (0,2√5).
What is y-intercept of a function?The intersection of the graph of the function with the y-axis gives y-intercept of that function. The y-intercept is the value of y on the y-axis at which the considered function intersects it.
Assume that we've got: y = f(x)
At y-axis, we've got x = 0, so putting it will give us the y-intercept.
Thus, y-intercept of y = f(x) is y = f(0)
We are given that;
Point b= 2 root 5 units
Now,
Since point B is 2√5 units away from point A, we can draw two circles with radius 2√5 centered at point A. The two intersections of these circles with the gridlines will give us the two possible positions of point B.
For example, if point A has coordinates (0,0) and the gridlines are the x-axis and y-axis, then the two possible positions of point B are:
B1: (2√5,0) - the intersection of the circle with the positive x-axis
B2: (0,2√5) - the intersection of the circle with the positive y-axis
If the gridlines are different, then the positions of point B will be different.
Therefore, by the intercept the answer will be (2√5,0) and B2: (0,2√5).
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Pet shelters typically have special adoption events for black cats and dogs, since their adoption rate is lower. One specific pet shelter (which has only cats and dogs) has 40% cats and 60% dogs. Twenty percent of the cats are black, and 10% of the dogs are black. The adoption percentage for black cats is 5%, and for black dogs is 10%. Cats that are not black are equally likely to be adopted as not adopted, and this is the same for dogs that are not black. For (b)-(f), write the probability statement and then calculate your answer.
a) Draw the probability tree and label all events and probabilities in it.
b) What is the probability that a randomly chosen animal is not black, a dog, and is adopted?
c) What is the probability that a randomly chosen animal is black, a cat, and is not adopted?
d) What is the probability that a randomly chosen animal is adopted?
e) Suppose a randomly chosen animal is a cat, what is the probability that it is adopted?
f) Suppose a randomly chosen animal is adopted, what is the probability that it is a cat?
a) The pet shelter has 40% cats (20% black, 5% adopted), and 60% dogs (10% black, 10% adopted). 50% are adopted in non-black cats and dogs, b) 0.45, c) 0.19, d) 0.15, e) 0.375, f) 0.6.
a) The probability tree would seem like this:
P(Cat) = 40%
P(Black Cat) = 20%
P(Adopted Black Cat) = 5%
P(Not Adopted Black Cat) = 95%
P(Not Black Cat) = 80%
P(Adopted Not Black Cat) = 50%
P(Not Adopted Not Black Cat) = 50%
P(Dog) = 60%
P(Black Dog) = 10%
P(Adopted Black Dog) = 10%
P(Not Adopted Black Dog) = 90%
P(Not Black Dog) = 90%
P(Adopted Not Black Dog) = 50%
P(Not Adopted Not Black Dog) = 50%
b) P ( Adopted & Not Black Dog ) = P(Not Black Dog) x P(Adopted | Not Black Dog) = 0.90 * 0.50 = 0.45
c) P(Not Adopted & Black Cat) = P(Black Cat) x P(Not Adopted | Black Cat) = 0.20 * 0.95 = 0.19
d) P(Adopted) = P(Adopted Black Cat) + P(Adopted Not Black Cat) + P(Adopted Black Dog) + P(Adopted Not Black Dog) = 0.05 + 0.50 + 0.10 + 0.50 = 0.15
e) P(Adopted | Cat) = P(Adopted & Cat) / P(Cat) = (0.05 + 0.50) / 0.40 = 0.375
f) P(Cat | Adopted) = P(Cat & Adopted) / P(Adopted) = (0.05 + 0.50) / 0.15 = 0.6.
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Un cubo tiene un volumen de 8 Metro cúbicos .¿cual es la longitud de una arista del cubo ? Muestra cómo hallasteis tu respuesta
The edge of the cube is equal to 2 m³.
QuadrilateralsThere are different quadrilaterals, for example: square, rectangle, rhombus, trapezoid and parallelogram. Each type is defined accordingly to its length of sides and angles. For example, in a square, all angles are 90° and all sides present the same value.
A square is a 2D figure, when the square is associated with a 3D figure, the figure is called a cube.
The cube presents 6 faces with length (l), height (h) and width (w) equal. Thus, the cube presents congruent edges. The volume is given from the formula V=a³, where a is a congruent edge.
The exercise gives the volume of the cube 8m³. As presented previously, the edges of the cube are congruent and the formula by the volume is V=a³. Thus,
V=a³
8=a³
[tex]\sqrt[3]{a^3}[/tex]=[tex]\sqrt[3]{8}[/tex]
a=[tex]\sqrt[3]{2^3}[/tex]
a=2 m
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help me on this question plss
An airplane on a transatlantic flight took 2 hours 30 minutes to get form New York to its destination, a distance of 3,000 miles. To avoid a storm, however, the pilot went off his course, adding a distance of 600 miles to the flight. How fast did the plane travel?
A) 1440mph
B) 1461mph
C) 1480mph
D) 1466mph
E) 1380mph
Please Help
Answer:
We can use the formula:
distance = rate x time
The total distance traveled by the plane is 3,000 + 600 = 3,600 miles. The time it took to cover this distance is 2 hours and 30 minutes, or 2.5 hours. So we have:
3,600 = rate x 2.5
Solving for the rate, we get:
rate = 3,600 / 2.5 = 1440 mph
Therefore, the answer is (A) 1440mph
A
group of four students is performing an
experiment with salt. Each student must
add teaspoon of salt to a solution. The
group only has a -teaspoon measuring
poon. How many times will the group
Need to fill the measuring spoon in order
To perform the experiment?
The number of times the group need to fill the measuring spoon in order to perform the experiment is 12 times
What is an Equation?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.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Let the amount of teaspoon of salt added to the solution be = ( 3/8 ) of spoon
Let the number of students = 4 students
The group only has ( 1/8 ) teaspoon measuring spoon
Substituting the values in the equation , we get
So , the amount of teaspoon of salt added to the solution by one student = amount of teaspoon of salt added to the solution / ( 1/8 )
On simplifying the equation , we get
The amount of teaspoon of salt added to the solution by one student = ( 3/8 ) / ( 1/8 )
The amount of teaspoon of salt added to the solution by one student = 3 times
And , the number of times the salt is added by 4 students = 4 x 3 = 12 times
Hence , the number of times is 12
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A colony of bacteria doubles every 12 hours.
a. Write an exponential function that represents the population of the bacteria after t
hours?
b. If the colony had 400 bacteria after 4 days, how many bacteria were there?
a) The exponential function that represents the population of the bacteria after t hours is given as follows:
y = a(2)^(t/12). (a is the initial amount).
b) The initial amount is given as follows: a = 2.
How to model the exponential function?The standard definition of an exponential function is given as follows:
y = a(b)^(x/n).
In which:
a is the initial amount.b is the rate of change.n is the time needed for the rate of change.Considering that the colony doubles in 12 hours, the parameters b and n are given as follows:
b = 2, n = 12.
Hence the function is defined as follows:
y = a(2)^(t/12).
After 4 days = 96 hours, there are 400 bacteria, hence the initial amount is obtained as follows:
400 = a(2)^(96/12)
256a = 400
a = 400/256
a = 2. (rounding).
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A bakery sold 105 cupcakes in one day. The head baker predicted he would sell 85 cupcakes that day. What was the percent error of the baker's prediction?
There was 19% percent error of the baker's prediction.
What is a percent error?Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.
Given that, a bakery sold 105 cupcakes in one day, the head baker predicted he would sell 85 cupcakes that day.
We are asked to find the percent error of the baker's prediction,
Percent error = |expected value - exact value| / exact value × 100 %
The expected value is the prediction of the Chef = 85
The exact value = 105
Percent error = |85-105| / 105 × 100 %
= 20/105 × 100 %
= 19%
Hence, there was 19% percent error of the baker's prediction.
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Calculate the area of each triangle using two different methods. Figures are not drawn to scale.
Who ever give me the answer gets extra points
The areas of the triangles are as follows:
1. 52.5 ft²
2. 369 inches²
How to find the area of a triangle?The area of a triangle can be represented as follows:
area of a triangle = 1 / 2 bh
where
b = base of the triangleh = height of the triangleTherefore, let's find the areas of the triangle using the formula as follows:
1.
area of the triangle = 1 / 2 × 15 × 7
area of the triangle = 1 / 2 × 105
area of the triangle = 52.5 ft²
2.
area of the triangle = 1 / 2 × 41 × 18
area of the triangle = 1 / 2 × 738
area of the triangle = 369 inches²
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Jacy keeps track of the amount of average monthly rainfall in her hometown. She determines that the average monthly rainfall can be modeled by the function ...where ...represents the average monthly rainfall in centimeters and ... represents how many months have passed. If ... represents the average rainfall in July, in which months does Jacy’s hometown get at least 10.5 centimeters of rainfall? Show all of your algebraic reasoning to support your final answer.
Answer:
September, October, November
Step-by-step explanation:
When monthly rainfall in centimeters is represented by A(t) = 2.3sin(πt/6)+9.25 and A(0) represents July's rainfall, you want to know the months in which average rainfall is at least 10.5 cm.
InequalityWe want to find the values of t that make A(t) ≥ 10.5:
2.3sin(πt/6) +9.25 ≥ 10.5
2.3sin(πt/6) ≥ 1.25 . . . . . . subtract 9.25
sin(πt/6) ≥ 0.543478 . . . . divide by 2.3
πt/6 ≥ 0.574575 . . . . . . . take inverse sine
t ≥ 1.097
The sine function is symmetrical about π/2, so this also means solutions will be of the form ...
πt/6 ≤ π -0.574575
t ≤ 6 -1.097 ≈ 4.903
MonthsThe month numbers that will have rainfall at least 10.5 cm will fall in the range ...
1.097 ≤ t ≤ 4.903
t ∈ {2, 3, 4}
If July is month 0, then these months are September, October, November.
Jacy's hometown will get at least 10.5 cm of rain in September, October, and November.
__
Additional comment
The attachment confirms this result. We have shifted the rainfall function so it can use conventional month numbers. It shows months 9, 10, 11 have rainfall above 10.5 cm.
<95141404393>
Enter the correct answer in the box. The graph of a quadratic function is represented by the table. x f(x) 6 -2 7 4 8 6 9 4 10 -2 What is the equation of the function in vertex form? Substitute numerical values for a, h, and k.
The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6
What is function?Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.
here, we have,
The given values are
x, f(x)
6, -2
7, 4
8, 6
9, 4
10, -2
The equation of the function in vertex form is given as follows;
f(x) = a × (x - h)² + k
To find the values of a, h, and k, we proceed as follows;
When x = 6, f(x) = -2
We have;
-2 = a × (6 - h)² + k = (h²-12·h+36)·a + k.............(1)
When x = 7, f(x) = 4
We have;
4 = a × ( 7- h)² + k = (h²-14·h+49)·a + k...........(2)
When x = 8, f(x) = 6...........(3)
We have;
6 = a × ( 8- h)² + k
When x = 9, f(x) = 4.
We have;
4 = a × ( 9- h)² + k ..........(4)
When x = 10, f(x) = -2...........(5)
We have;
-2 = a × ( 10- h)² + k
Subtract equation (1) from (2)
4-2 = a × ( 7- h)² + k - (a × (6 - h)² + k ) = 13·a - 2·a·h........(6)
Subtract equation (4) from (2)
a × ( 9- h)² + k - a × ( 7- h)² + k
32a -4ah = 0
4h = 32
h = 32/4
= 8
From equation (6) we have;
13·a - 2·a·8 = 6
-3a = 6
a = -2
From equation (1), we have;
-2 = -2 × ( 10- 8)² + k
-2 = -8 + k
k = 6
The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6
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What is the product of 2.5\times 10^22.5×10
2
and 3.7 \times 10^53.7×10
5
expressed in scientific notation?
The solution is, the product is 9.25* 10^83.2.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
here, we have,
2.5\times 10^22.5×10^2 and 3.7 \times 10^53.7×10^5
=2.5* 10^22.5×10^2 × 3.7 * 10^53.7×10^5
=9.25* 10^24.5*10^58.7
=9.25* 10^83.2
Hence, The solution is, the product is 9.25* 10^83.2.
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Find the equation to the plane through the point (−1,3,2) and perpendicular to the planes x+2y+2z=11 and 3x+3y+2z=15.
The equation of the plane that passes through (-1, 3, 2) and is perpendicular to the planes x + 2y + 2z = 11 and 3x + 3y + 2z = 15 is -2x + 4y - 3z - 17 = 0
In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. A plane can be defined by an equation in the form of ax + by + cz = d, where a, b, and c are coefficients that determine the plane's orientation and d is a constant.
To find the equation of a plane, we need to know its normal vector, which is a vector perpendicular to the plane. We can find the normal vector of the plane we are looking for by taking the cross product of the normal vectors of the two given planes.
The normal vector of the first plane, x + 2y + 2z = 11, is (1, 2, 2), and the normal vector of the second plane, 3x + 3y + 2z = 15, is (3, 3, 2). To take their cross product, we can use the following formula:
n = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
where n is the normal vector, and a and b are the two given normal vectors. Plugging in the values, we get:
n = (2(2) - 2(3), 2(3) - 1(2), 1(3) - 2(3)) = (-2, 4, -3)
This means that the normal vector of the plane we are looking for is (-2, 4, -3). We also know that the plane passes through the point (-1, 3, 2), so we can use the point-normal form of the equation of a plane, which is:
a(x - x₀) + b(y - y₀) + c(z - z₀) = 0
where (x₀, y₀, z₀) is the given point, and a, b, and c are the coefficients of the normal vector. Plugging in the values, we get:
-2(x + 1) + 4(y - 3) - 3(z - 2) = 0
Simplifying, we get:
-2x + 4y - 3z - 17 = 0
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can you please solve?
The equation of the line is y = (2/3)x + 20/3.
What is an equation of a line?The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
Example:
The slope of the line y = 2x + 3 is 2.
The slope of a line that passes through (1, 2) and (2, 3) is 1.
We have,
The equation has:
Slope = 2/3
Passes through the line (-4, 4).
Now,
The equation of the line is y = mx + c.
m = 2/3
Consider (-4, 4) = (x, y)
So,
4 = (2/3)(-4) + c
4 = -8/3 + c
c = 4 + 8/3
c = (12 + 8) / 3
c = 20/3
Now,
y = (2/3)x + 20/3
Thus,
The equation of the line is y = (2/3)x + 20/3.
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Which properties are true for all parallelograms?
The requried properties are true for all the parallelograms are,
1. Opposite sides are parallel 2. Opposite sides are congruent
A parallelogram is a quadrilateral consisting of pairs of parallel sides.
The following properties are true for all parallelograms,
The opposite sides are parallel.
The Opposite sides are congruent.
All parallelograms have four congruent sides (property 1), four right angles (property 2), or exactly one pair of parallel sides (property 3). These properties only apply to specific types of parallelograms, such as rectangles or squares.
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What is the slope of the line?
Answer:
The slope is 3.
Step-by-step explanation:
You can find this by using rise/run.
The line goes from (1,2) to (2,5).
The rise for this is 3 and the run is 1.
3/1 is 3, therefore the slope is 3
A student attaches a 3.0 kg mass to a spring with a spring constant of 40 N/m. The student compresses the spring to the left by 0.15 m and then releases the mass allowing it to oscillate. Which of the following equations best describes the position vs. time relationship?
A. x = (3.0 kg) cos (1.42 t)
B. x = (-15 m) cos (1.42 t)
C. x = (-40 N/m) cos (2πt)
D. x = (-0.15 m) cos (3.65 t)
The position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .
From the given data,
A student attaches a 3.0 kg mass to a spring with a spring constant of 40 N/m.
The student compresses the spring to the left by 0.15 m and then releases the mass allowing it to oscillate.
The position of the mass for the spring- mass oscillatory system is,
[tex]$x=A \cos \omega t$[/tex]Where
ω = angular frequency of the wave.T = time period of the wave.The angular frequency of the system is,
The angular frequency refers to the angular displacement of any wave element per unit of time or the rate of change of the waveform phase.
[tex]$$\omega=\sqrt{\frac{k}{m}}$$[/tex]Therefore, equation (1) changes as follows
[tex]x & =A \cos \left(\sqrt{\frac{k}{m}}\right) t \\[/tex]
[tex]& =-(0.15 \mathrm{~m}) \cos \left(\sqrt{\frac{40 \mathrm{~N} / \mathrm{m}}{3.0 \mathrm{~kg}}}\right) t \\[/tex]
[tex]& =-(0.15 \mathrm{~m}) \cos (3.65) t[/tex]
Therefore, the position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .
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Find each value or measure. Assume that segments that appear to be tangent are tangent.
The measure of arc AD is 25°.
AC and BD are chords of the circle. The two chords intersect at the point E which makes an angle 93°. The measure of arc BC is 161°. To find the measure of arc AD:
The measure of arc AD by using the property that "if two chords intersect in the interior of the circle, then the measure of each angle is half the sum of the arcs intercepted by the angles and its vertical angle".
Thus, applying the above theorem:
93° = 1 /2 (arcBC + arcAD)
Here, measure of angle E is 93°.
93° = 1 / 2( 161 + AD)
93 = 161 / 2 + AD / 2
93 = 161/ 2 + AD/2
solving further ( multiplying equation by 2 on both sides)
186 = AD + 161
AD = 186 - 161
arc AD = 25°
So, the measure of arc AD is 25°.
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____The given question is incomplete, the complete question is given below:
Find each value or measure. Assume that segments that appear to be tangent are tangent. Find arc AD.
Translate this sentence into an equation.
Chau's savings increased by 17 is 54.
Answer: 17x = 54
x represents chau's savings before it increased
1. 3mm : 1cm= what is the ratio
The equivalent ratio to the given expression 3mm : 1cm as required in the task content is; 3 : 10.
What is the equivalent ratio of 3mm : 1cm?It follows from the task content that the equivalent ratio of 3mm : 1cm is to be determined .
Recall, 1cm is equivalent to 10 mm.
On this note, the given ratio can be written as; 3mm : 10 mm.
Consequently, the ratio which is equivalent to the given expression is; 3 : 10.
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The line segment joining the points P(-3,2) and Q(5,7) is divided by the y-axis in the ratio:
Answer:
Step-by-step explanation:
The line segment joining two points P and Q can be represented by the equation of a straight line in the form y = mx + b, where m is the slope and b is the y-intercept.
To find the equation of the line, we need to find the slope, which can be calculated using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the points P and Q, respectively.
In this case, the coordinates are:
P = (-3, 2) and Q = (5, 7)
So, the slope is:
m = (7 - 2) / (5 - (-3)) = 5 / 8
Next, we can use either of the points to find the y-intercept. Let's use point P:
b = y - mx, where y and x are the y and x coordinate of the point, respectively.
In this case,
b = 2 - m * (-3) = 2 - (5/8) * (-3) = 2 + 15/8 = 89/8
So, the equation of the line joining the points P and Q is:
y = (5/8)x + 89/8
Now, to find the point where the line crosses the y-axis, we need to find the x-coordinate of the point where y = 0.
So, we have:
0 = (5/8)x + 89/8
Solving for x, we get:
x = -(89/8) / (5/8) = -89 / 5
This means that the line crosses the y-axis at the point (-89/5, 0). To find the ratio in which the line segment is divided by the y-axis, we need to find the ratio of the distance from the y-axis to point P to the distance from the y-axis to point Q.
Let's call the point of intersection with the y-axis R. The distances are then:
PR = (3, 2) and QR = (5 - (-89/5), 7)
The ratio of the distances is then:
PR / QR = (3, 2) / (5 - (-89/5), 7) = 3 / (5 + 89/5) = 3 / (94/5) = 15/47
So, the line segment joining the points P and Q is divided by the y-axis in the ratio 15:47.
Problem 6. Determine if the pair of planes is parallel or intersecting. If they are parallel, find the distance between them. If they intersect, find a parametrization for the line of intersection. 1. 3.x + 2y +z = 8 and x + 3y – z = 5. 2. 2.c – y +z = 4 and 4.0 – 2y + 2z = 3.
1. To determine if the pair of planes 3x + 2y + z = 8 and x + 3y – z = 5 are parallel or intersecting, we can check if their normal vectors are parallel or not.
The normal vector of the first plane is <3, 2, 1>, and the normal vector of the second plane is <1, 3, -1>. These normal vectors are not parallel, so the planes intersect at a line.
To find a parametrization for the line of intersection, we can set one of the variables to be a parameter and solve for the other two. Let's choose z as the parameter.
From the second plane equation, we have z = x + 3y - 5. Substituting this into the first plane equation, we get: 3x + 2y + (x + 3y - 5) = 8
Simplifying, we get: 4x + 5y = 13
Solving for y in terms of x, we get: y = (13 - 4x) / 5
So a parametrization for the line of intersection is: x = t
y = (13 - 4t) / 5, z = t + 3((13 - 4t) / 5) - 5 = (2t + 2) / 5
2. To determine if the pair of planes 2c – y + z = 4 and 4c – 2y + 2z = 3 are parallel or intersecting, we can again check if their normal vectors are parallel or not.
The normal vector of the first plane is <2, -1, 1>, and the normal vector of the second plane is <4, -2, 2>. These normal vectors are parallel (in fact, they are scalar multiples of each other), so the planes are parallel.
To find the distance between the two parallel planes, we can take any point on one plane and find its distance to the other plane. Let's choose the point (0, 0, 4) on the first plane. The distance between this point and the second plane is: |4c - 2(0) + 2(4) - 3| / sqrt(4^2 + (-2)^2 + 2^2) = |4c - 5| / 2sqrt(6)
So the distance between the two parallel planes is |4c - 5| / 2sqrt(6).
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Write two fractions that are equivalent to 4/6
Answer:
2/3, 8/12
Step-by-step explanation:
(4 divided by 2)/(6 divided by 2) = 2/3
(4 times 2)/(6 times 2)= 8/12
Multiply/Divide the numerator and the denominator by the same number
There are infinte equivalent fractions to 4/6! 4000/6000 is one!
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Richard used a radius measure of a circle to be 3.6 inches when he calculated the area of a circle. The correct radius measure was actually 3.5 inches. What is the difference between Richard’s measured area of the circle and the actual area of the circle?
a. 0.1π square inches
b. 0.71π square inches
c. π square inches
d. 0 square inches
e. 2.4 square inches
As a result, the answer is (e) 2.4 square inches, which is the difference between Richard's measured and real circle area.
What is area?In mathematics, area is a measure of the amount of space occupied by a two-dimensional object, such as a rectangle, triangle, circle, or any other shape. It's a scalar quantity that describes the size of a region in two-dimensional space. The units of area are typically square units, such as square inches, square centimeters, square meters, etc. In general, the area of a shape is a measure of how much space it occupies, and it is an important concept in geometry, engineering, and many other fields.
Here,
The formula for the area of a circle is given by:
A = πr²
Where r is the radius of the circle.
Using the incorrect radius of 3.6 inches, the calculated area would be:
A = π * (3.6 inches)² = 40.44 square inches
Using the correct radius of 3.5 inches, the actual area would be:
A = π * (3.5 inches)² = 38.5 square inches
So, the difference between Richard's measured area of the circle and the actual area of the circle would be:
40.44 square inches - 38.5 square inches = 1.94 square inches
Therefore, the answer is (e) 2.4 square inches that is the difference between Richard’s measured area of the circle and the actual area of the circle.
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