An arithmetic sequence (AP) is a progression in which the difference (called the tolerance) between two consecutive terms in the passage/progression is constant.
Therefore, the expression for the nth term is,
an = a₁ + (n − 1) d
and in general
an = am + (n - m) d.
where d is the difference.
For an arithmetic series with initial term a₁ and difference between terms of d
an = a₁ + (n − 1) d
We are given some of the values,
d = 6 and
S8 = 440
So, we will put the values in the formula and calculate the desired value,
a1 = 440 = a1 + ( 8 - 1 ) * 6 = a1 + 42
a1 = 440 - 42 = 398.
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In an arithmetic sequence with a₁ =4 and d=9 , which term is 184 ?
The value of 184 is 21th term value.
How to find the which term is 184?The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same value.
The formula of Arithmetic progression is
a(n) = a1 + (n - 1)d
given that the terms of arithmetic sequence,
a2 = 4 and d = 9.
now find the which term is 184.
a(n) = a1 + (n - 1)d
now, substitute a(n) = 184, d = 9.
184 = 4 + (n - 1)9
184 = 4 + 9n -9
184 = -5 +9n
184 + 5 = 9n
n = 189/9
n = 21
Hence,the value of 184 is 21th term value.
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of 6
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The distribution of weight for 9-ounce bags of a particular brand of potato chips can be modeled by a normal distribution with
mean = 9.12 ounces and standard deviation o=0.05 ounce. Sketch the normal density curve. Label the mean and the points
that are 1, 2, and 3 standard deviations from the mean. Do not round your answers.
The graph of the normal distribution is given at the end of the answer.
What does the Empirical Rule state?It states that, for a normally distributed random variable, approximately:
68% of the values in the distribution are within 1 standard deviation of the mean.95% of the values in the distribution are within 2 standard deviations of the mean.99.7% of the values in the distribution are within 3 standard deviations of the mean.The normal distribution is symmetric, hence, from the Empirical Rule, we get that most observations will be clustered around the mean, which is the center of the distribution.
The mean is of 9.12 and the standard deviation is of 0.05, hence:
68% of the values are between 9.07 and 9.17.95% of the values are between 9.02 and 9.22.99.7% of the values are between 8.97 and 9.27.The graph of the distribution is given at the end of the answer.
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Find f(-3), f(0) , and f(1) for the function f(x) = (2/5)x-2
The f(-3), f(0) , and f(1) is (-16/5), -2 and (-8/5).
The given function is f(x) = (2/5)x-2
1) f(-3) = x = -3
Substituting the value of x in the equation f(x) = (2/5)x - 2, we get -
f(-3) = (2/5)x - 2
f(-3) = (2/5)(-3) - 2
f(-3) = (-6/5) - 2
f(-3) = (-6 -10)/5
f(-3) = -16/5
2) f(0) = x = 0
Substituting the value of x in the equation f(x) = (2/5)x - 2, we get -
f(0) = (2/5)(0) - 2
f(0) = -2
3) f(1) = x = 1
Substituting the value fo x in the equation f(x) = (2/5)x - 2, we get -
f(1) = (2/5)(1) - 2
f(1) = (2/5) - 2
f(1) = (2 - 10)/5
f(1) = -8/5
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ARE YOU GOOD AT ALGEBRA? PLS HELP. INDICATE ALL TRANSFORMATIONS AND JUSTIFY EACH ONE!
Answer:
See explanation
Step-by-step explanation:
f(x)=|x|
f(x) ==> f(x+3): translation of function 3 units to the left, f(x-3) would be translation 3 units right
f(x+3) ==> 3f(x+3): vertical stretch by a factor of 3
f(x+3) ==> -3f(x+3): Reflection over the x-axis, since y value changes over the x-axis while x value changes over the y-axis
-3f(x+3) ==> h(x)=-3f(x+3)+1: Translation one unit up.
y varies directly with x .
If y=7 when x=2 , find x when y=3 .
The required value of x is 6 / 7.
What is the constant of proportion?
A constant of proportion is illustrated as the ratio which relates two given quantities with each other in the relationship of proportion. Constant of proportion is also called constant variation, rate of change, and constant ratio.
Solving for the value of y
let k be the constant of proportion
When a quantity varies directly with others, then we have a relation, y = kx —-- 1
Now, find constant of proportion for given x = 2 and y = 7, we need to put these values in equation 1, i.e.
7 = k(2)
k = 7 / 2
Further calculate x for given y = 3 i.e.
3 = (7 / 2)x
x = 6 / 7
Hence, the required value of x is 6 / 7.
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HELP ASAP!!!!!!!
Which expression is equivalent to −5 + 8 − 6x − 8x?
1.) 14x + 13
2.)-14x - 3
3.)-14x + 3
4.)-14x + 13
The expression 3-14x is equivalent to -5+8-6x-8x.
The phrase has the same meaning as,
=-5+8-6x-8x
=(3)-(14x)
=3-14x
Any combination of terms that have undergone operations like addition, subtraction, multiplication, division, etc. is known variable expression. Let's use the equation 5x + 7 as an example. Thus, 5x + 7 is an illustration of an algebraic expression.
There are three primary categories of algebraic expressions, including: Numeral Expression. Binary Expression of a polynomial.
Properties:
Equivalence Property.Associative Quality.Discretionary PropertyProperty of identity.Contrary Property.Hence, the expression 3-14x is equivalent to -5+8-6x-8x.
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Answer: -14x + 13
Step-by-step explanation:
A football is punted into the air. Its height h, in meters, after t seconds is given by the equation h=-4.9t²+25.5t+1. Round your answer to hundredths place if needed.
1. How high is the ball after 3 seconds?
2. What is the maximum height of the ball and how many seconds will it take to reach that point?
3. How long will the ball be in the air?
Using the given quadratic function, it is found that:
1. The ball has a height of 33.4 meters after 3 seconds.
2. The maximum height of the ball is of 34.18 meters, taking 2.6 seconds to reach it's height.
3. The ball will be in the air for 5.24 seconds.
What is the quadratic equation for the ball's height?
The quadratic equation for the ball's height after t seconds is given by:
h(t) = -4.9t²+ 25.5t + 1.
Hence, after 3 seconds, the height is:
h(3) = -4.9(3)² + 25.5(3) + 1 = 33.4 meters.
The ball has a height of 33.4 meters after 3 seconds.
What is the maximum height of the ball?To find the maximum height of the ball, we have to find the vertex of the function.
What is the vertex of a quadratic equation?A quadratic equation is modeled by:
y = ax^2 + bx + c
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
[tex]x_v = -\frac{b}{2a}[/tex][tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]Considering the coefficient a, we have that:
If a < 0, the vertex is a maximum point.If a > 0, the vertex is a minimum point.For this problem, the coefficients are given as follows:
a = -4.9, b = 25.5, c = 1.
Hence:
[tex]x_v = -\frac{25.5}{2(-4.9)} = 2.6[/tex][tex]y_v = -\frac{(25.5)^2 - 4(-4.9)(1)}{4(-4.9)} = 34.18[/tex]The maximum height of the ball is of 34.18 meters, taking 2.6 seconds to reach it's height.
How long the ball was in the air?To find how long the ball was in the air, we have to solve the quadratic equation, hence:
[tex]\Delta = 25.5^2-4(-4.9)(1) = 669.85[/tex][tex]t_1 = \frac{-25.5 + \sqrt{669.85}}{2(-4.9)} = -0.04[/tex][tex]t_2 = \frac{-25.5 - \sqrt{669.85}}{2(-4.9)} = 5.24[/tex]Time has to assume positive values, hence:
The ball will be in the air for 5.24 seconds.
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A flea landed on a coordinate grid. The flea hopped across the x -axis and then across the y -axis in the form of two consecutive reflections. Then it walked 9 units to the right and 4 units down. If the flea's final position was at (4,-1) , what point did it originally land on?
The original coordinates of flea were ( 5 , - 3) .
What does coordinate mean in this case?
To coordinate is to make plans for things to come together or to collaborate with another person to achieve a goal. Coordination occurs when all the details of a journey are planned. When you and a friend divide up a large science project and decide who will do what, that is an example of coordination.F' = ( 4 , - 1 )
To undo translation of 9 units’ right and 4 units down, translate 9 units left and 4 units up
( 4, - 1 ) → ( 4 - 9 , -1 + 4 )
→ ( -5,3)
Reflect back across Y-axis to undo reflection means change x -coordinate by -1
(- 5,3) → ( 5, 3)
Reflect back across x-axis to undo reflection means change y-coordinate by -1
( 5 , 3) → ( 5 , - 3)
Therefore the original coordinates of flea were ( 5 , - 3)
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The rectangle shown has an area of 84 square inches. Find the dimensions of the rectangle.
The dimensions of the rectangle:
Length = 12 inches
Width = 7 inches
What is the Area of a Rectangle?Area of a rectangle = (length)(width)
Given the following:
Area of the rectangle = 84 square inches
Length = x + 3
Width = x - 2
Therefore:
(x + 3)(x - 2) = 84
Expand
x² + x - 6 = 84
x² + x - 6 - 84 = 0
x² + x - 90 = 0
Factorize
(x - 9)(x + 10) = 0
x = 9 or x = -10
Substitute x = 9 into each expression to find the dimensions of the rectangle:
Length = x + 3 = 9 + 3 = 12 inches
Width = x - 2 = 9 - 2 = 7 inches
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Can Pi be written as a fraction?
Please help its my last question on the test and its due today!
i will give "Thanks" and "Brainlist"
Answer:
Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction.
Answer:
No
Step-by-step explanation:
It can not be a fraction because it is a never-ending/infinite decimal.
A school system is reducing the amount of dumpster loads of trash removed each week. in week 5, there were 45 dumpster loads of waste removed. in week 10, there were 30 dumpster loads removed. assume that the reduction in the amount of waste each week is linear. write an equation in function form to show the amount of trash removed each week. f(x) = −3x 45 f(x) = 3x 45 f(x) = −3x 60 f(x) = 3x 60
f(x) = −3x + 60 is the linear equation in function form to show the amount of trash removed each week.
Let x be the number of weeks and y be the trash removed
Then , the linear equation can be given as
y=mx+c
To find out the slope and intercept, let's have a look at the given conditions.
For week 5, we can write the equation as
45 = 5m +c ... (1)
For week 10, we can write the equation as
30 =10m +c ......(2)
Solving both the equation by elimination method
(45 = 5m +c) * 2
30 = 10m+c
c = 60
Now, finding the value of x by substituting value of c in eq (1)
45 = 5m + 60
-15 = 5m
m = -3
Now, the required linear equation becomes
y = -3x + 60
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Find reference angle and exact value
The reference angle of the angle of -240º is of 60º.
The sine of the angle is given by:
[tex]\sin{-240^\circ} = \frac{\sqrt{3}}{2}[/tex]
How to find the reference angle of -240º?The first step towards finding the equivalent angle of a negative angle is finding it's positive equivalent, adding 360º to it, hence:
-240º + 360º = 120º.
Then we have to find the equivalent on the first quadrant. 120º is on the second quadrant, hence:
180º - 120º = 60º.
The reference angle of the angle of -240º is of 60º.
On the second quadrant, the sine is positive, hence:
[tex]\sin{-240^\circ} = \sin{60^\circ} = \frac{\sqrt{3}}{2}[/tex]
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Determine whether the statement is always, sometimes, or never true. Explain you reasoning.
If point A lies in plane P , then AB↔ lies in plane P.
A line segment is also a line but a line with finite length and fixed endpoints. The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true.
What is a line?A line is a one-dimensional shape that is straight, has no thickness, and extends in both directions indefinitely.
A line segment is also a line but a line with finite length and fixed endpoints.
The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true. Because if Point A and B both lie on the same plane, p then line AB will lie on plane P. Therefore, the given statement "If point A lies in plane P, then AB↔ lies in plane P." will be true.
But if Point A lies on plane P, while point B lies on another plane, then line AB will not lie on plane P. Therefore, the given statement "If point A lies in plane P, then AB↔ lies in plane P." will be false.
Hence, The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true.
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Could I get an equation for this problem please?
*See picture for word problem
The number of tacos that you can get if a drink cost $1.50 and each taco cost $.75 is 4.67.
Word problemCost of a drink = $1.50Cost of each taco = $.75Total amount with you = $5Number of tacos bought = t5 = 1.50 + 0.75t
subtract 1.50 from both sides5 - 1.50 = 0.75t
3.50 = 0.75t
divide both sides by 0.75t = 3.50 / 0.75
t = 4.666666666666666
Approximately,
t = 4.67
Therefore, the number of tacos that you can get if a drink cost $1.50 and each taco cost $.75 is 4.67
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x < -3
Solutions:
X=
X=
X=
On a Number Line
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
The values of the x in number line are -7 -6 -5 -4.
According to the statement
We have to find that the numbers in the number line.
So, For this purpose, we know that the
A number line is a horizontal line that has equally spread number increments
From the given information:
The given equation is a x < -3 And a numbers givens On a Number Line are:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
It means we have to find the numbers in a number line less than the -3.
we know that the large numbers with the negative sign are a smaller number and
the Small numbers with the negative sign are a larger number.
According to this,
The numbers -7 -6 -5 -4 are the values of the x less than the -3.
So, The values of the x in number line are -7 -6 -5 -4.
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Identify all the correct names for the given angle.
Answer: id k if this was what you were looking for but i tried
Step-by-step explanation:
h=acute
rhp=acute
vhp=acute
v=straight angle
phv=acute
Find the slope of the line through each pair of points. (-2,-4) and (1,2)
The slope of the line passing through the pair of points, (-2 , -4) and (1 , 2), is 2.
The slope, m, determines the steepness of a line. It can be measured by the vertical distance from one point to another divided by the horizontal distance of the same points. The formula for solving the slope is:
m = (y2 - y1)/(x2 - x1)
where m is the slope
x1 and y1 are the coordinates of one point
x2 and y2 are the coordinates of the 2nd point
Given two points in the line, (-2 , -4) and (1 , 2), plug in the values to the formula for the slope.
m = (y2 - y1)/(x2 - x1)
m = (2 - -4)/(1 - -2)
m = 6/3
m = 2
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what is the answer to
-2(2x+3)= -4(x+1)-2
Step-by-step explanation:
-4x-6=-4x-4-2
-4x+4x=6-4-2
x=1
45, -72, and 0 all are examples of:
Positive integers
Negative integers
Irrational numbers
Integers
Answer:
Integers
Step-by-step explanation:
They are not all positive integers because -72 is negative.
They are not all negative because 45 is positive. The + sign is implied.
They are all whole numbers and not irrational.
Thus, they are integers, which by looking at the data, we see is true.
The endpoints of are cap A open paren negative 8 comma 5 close paren(−8, 5) and cap B open paren 0 comma 7 close(0, 7) . Find the coordinates of the midpoint cap m
The coordinates of the midpoint M of the segment AB are; (-4, 6).
What are the coordinates of the midpoint AB?It follows from the task content that points A and B have coordinates; (-8,5) and (0,7) respectively.
Since Midpoint is given as; {(x1 +x2)/2, (y1 +y2)/2}.
It follows that the midpoint in this scenario is;
= {(-8 +0)/2, (5 +7)/2}
= (-4, 6).
Ultimately, the midpoint is; (-4, 6).
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Given h(x) = -x + 5, find h(-5).
Two spheres have surface areas of 100 \pi square centimeters and 16π square centimeters. What is the ratio of the volume of the large sphere to the volume of the small sphere?
The ratio of the volume of the large sphere to the volume of the small sphere is 125/8.
Given two spheres having surface areas of 100π square centimeters and 16π square centimeters, solve for the radius of each spheres using the formula for the surface area of sphere.
A = 4πr^2
Sphere 1 : 100π cm^2 = 4πr^2
r^2 = 25
r = 5 cm
Sphere 2 : 16π cm^2 = 4πr^2
r^2 = 4
r = 2 cm
Then, solve for the volume of each spheres using the formula:
V = 4/3πr^3
Sphere 1 : V = 4/3π(5cm)^3
V = 500/3 π cm^3
Sphere 2 : V = 4/3π(2cm)^3
V = 32/3 π cm^3
Getting the ratio of the volume of the large sphere to the volume of the small sphere,
ratio = 500/3 π cm^3 / 32/3 π cm^3
ratio = 125/8
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...Sorry................
The Average consumption per person in the country in the given year is 7.6 gallons
How to find average consumption?
Average is found by adding the several amounts together and then dividing by the total number of amounts added.
Given that the annual consumption of fruit juice of a particular country = the total consumption by whole population in that year= [tex]2.28*10^{9}[/tex] gallons
Total population of that country in the given year = [tex]3.0*10^{8}[/tex]
Therefore Average consumption per person
=total consumption / total population
[tex]\frac{2.28*10^{9} }{3.0*10^{8} } \\=0.76*10\\=7.6[/tex]
This means the Average number of gallons consumed per person in the country in the given year is 7.6 gallons
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Paige solved the inequality -1/2x > -6 by using the Division Property of Inequality to get x >12 Did Paige solve the inequality correctly?
Using the Division Property of Inequality, Paige was able to get x > 12 and solve the inequality -1/2x > -6. Paige solve the inequality correctly.
Given that,
Using the Division Property of Inequality, Paige was able to get x > 12 and solve the inequality -1/2x > -6.
According to the division property of inequality, if we divide an equation's two sides by the same number, the equation doesn't change.
Let's say you have two identically sized pizzas. You divide each one into halves and serve your buddies one half of each pizza. You will have equal pieces of both pizzas left over if you measure the remaining portions.
-1/2x>-6
-1x>-6[tex]\times[/tex]2
-1x>-12
multiplying -1 on both sides we get
x>12
Therefore, Paige solve the inequality correctly.
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COORDINATE GEOMETRY Find the distance from P to l. Line l contains points (-3,0) and (3,0) . Point P has coordinates (4,3) .
According to the given statement the perpendicular distance form the point (4,3) is 3.
What Exactly Is Coordinate Geometry?Coordinate Geometry can be used to define points in space. The primary axis labeled a x- and y-axes is defined first, and then the points were measured and labelled in relation to these points. Furthermore, numerous geometric forms such as a line, curve, circle, ellipse, and hyperbola can be plotted inside the coordinate axes and their attributes studied.
According to the given data:the equation for line ℓ.
Begin by finding the slope of the line through the given points, (-3,0) and (3,0):
m = (Y₂ - Y₁)/(x₂ - x₁)
= (0 - 0)/(-3 - (-3))
= 0/0
The equation of the line using point (3,0) on the line:
Y - Y₁ = m(x - x₁)
putting value we get: (m = 0)
Y =0(x + x₁)
Y = 0
The equation of the line w perpendicular to line ℓ through P(4,3).
The slop of line l is 0.
hence this will be x- axis
and perpendicular distance from a point (x1,y1) to x- axis will be y1.
So the perpendicular distance form the point (4,3) is 3.
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The sides of a triangle have lengths 2 x, 8 , and 12. If the length of the longest side is 2 x , what values of x make the triangle acute?
The values of x to make the triangle acute is [tex]2 < x < 2\sqrt{13}[/tex].
Triangle inequality theorem states that that the sum of any two sides of the triangle is greater than the third side of of the triangle
Pythagorean Inequality theorem states that if the square of the longest side is less than the sum of squares of other two sides then the triangle is acute.
In the given question
Longest side = 2x
other two shorter side = 8 and 12.
By Triangle Inequality theorem
(i) 8+12>2x ⇒20>2x ⇒x<10
(ii) 2x+8>12 ⇒2x>4 ⇒x>2
(iii) 2x+12>8 ⇒2x>-4 ⇒x>-2
From (I), (ii) & (iii) we conclude that value of x should be in between 2<x<10.
To make the triangle acute
Using Pythagorean Inequality theorem
[tex](2x)^{2} < 8^{2} +12^{2} \\ \\ 4x^{2} < 64+144\\ \\ 4x^{2} < 208\\ \\ x^{2} < \frac{208}{4}[/tex]
[tex]x^{2} < 52\\ \\ taking. Square. Root . both . the . sides \\ \\ x < \sqrt{52} \\ \\ x < 2\sqrt{13}[/tex]
Therefore , the value of x to make the triangle acute is [tex]2 < x < 2\sqrt{13}[/tex].
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write 3 names for the angle
The names of the three angles are:
∠ABC = acute angle
∠FGH = right angle
∠QRS = obtuse angle
We have,
From the figure given,
The angles must be either right angles, acute angles, or obtuse angle.
Acute Angle: An acute angle is an angle that measures less than 90 degrees. It is smaller than a right angle.
Obtuse Angle: An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. It is larger than a right angle.
Right Angle: A right angle is an angle that measures exactly 90 degrees. It forms a perfect "L" shape and is typically denoted by a square corner symbol (∟).
Now,
The angle is less than 90 degrees.
∠ABC = acute angle
The angle is 90 degrees.
∠FGH = right angle
The angle is greater than 90 degrees.
∠QRS = obtuse angle
Thus,
The names of the three angles are:
∠ABC = acute angle
∠FGH = right angle
∠QRS = obtuse angle
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Simplify the expression
3\sqrt{27x^6y^4}
The simplified form of given expression 3√{27x^6y^4} is x³y²9√3
In this question, we have been given an expression 3\sqrt{27x^6y^4}
We need to simplify given expression.
3√{27x^6y^4}
= 3√{(3 × 9) x^6 y^4}
= 3√(3 × 9) × √x^6 × √y^4
= 3 (3√3) × x^(6/2) × y^(4/2)
= 9√3 × x³ × y²
= x³y²9√3
Therefore, the simplified form of given expression 3√{27x^6y^4} is x³y²9√3
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An object is moving at a speed of 1 yard every 7.5 months. Express this speed in centimeters per hour. Round your answer to the nearest hundredth.
*Note: you must use these exact conversion factors to get this question right.
Based on the speed of the object at 1 yard every 7.5 months, the speed in centimeters per hour is 0.01693 cm/ hour
What is the speed of the object?First, convert 1 yard to centimeters:
1 yard = 91.44 cm
Convert 7.5 months to hours:
= 7.5 months x 30 days per month x 24 hours per day
= 5,400 hours
The speed in centimeters per hour can therefore be found to be:
= 91.44 / 5,400
= 0.01693
= 0.01693 cm/ hour
In conclusion, the object is moving at a speed of 0.01693 cm/ hour
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Construct an equilateral triangle. Verify your construction using measurement and justify it using mathematics. (Hint: Use the construction for copying a segment.)
A triangle PRS is an equilateral triangle.
An equilateral triangle is a triangle whose all the three sides are equal and each angle measures 60º.
To construct an equilateral triangle:
1) First draw a ray PQ.
2) Then, with P as center and radius equal to 5 cm, draw an arc to cut ray PQ at R.
⇒ PR = 5 cm
3) Then, with R as the center and radius equal to 5 cm, draw an arc to intersect the initial arc at S.
4) Join PS and RS.
Now, the triangle PRS is an equilateral triangle with each side length of 5 cm.
Now mathematically we prove that the triangle we've constructed is an equilateral triangle.
PR = PS .......................(the radius of the arc is same)
PR = RS .......................(the radius of the arc is same)
Hence by transitivity property,
PR = RS = PS which is 5 cm
Therefore, triangle PRS is an equilateral triangle.
Learn more about an equilateral triangle here:
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