A Ferris wheel with a diameter of 10 m and makes one complete revolution every 80
seconds. Assume that at time t = 0, the Ferris Wheel is at its lowest height above
the ground of 2 m. You will develop the equation of a cosine graph that models your
height, in metres, above the ground as you travel on the Ferris Wheel over time, t in
seconds. To do this, answer the following questions.
1. State the amplitude of the graph.
2. State the value of k in the general form y = a cos [k(x − d)] + c.
-
3. State the value of d.
4. State the value of c.
5. State the cosine equation of the graph.
Answers
A Ferris wheel with a diameter of 10 m and makes one complete revolution every 80 seconds. Assuming that at time t = 0, the Ferris Wheel is at its lowest height above the ground of 2 m, the cosine equation of the graph drawn is, y = 5 cos [( π/40)(x - (π/2))] + 3. Here, amplitude of the graph is 5, value of k is π/40, d is π/2 and c is 3.
Developing the Equation of a Cosine Graph
The given information constitutes the following,
Diameter = 10 m
⇒ Radius, r = 5 m
Time, t = 80 s
Height above the ground, h = 2 m
Thus, we can infer that,
Amplitude, A = 5 m
Period, T = 80 s
Minimum height = 2 m
The cosine function is given as,
a cos [k(x − d)] + c
Here, A is amplitude
B is cycles from 0 to 2π and thus period = 2π/k
d is horizontal shift
c is vertical shift (displacement)
Now, 2π/k = 80
⇒ k = 2π/80 = π/40
The value of c is given as,
c = Amplitude - Minimum height
c = 5 - 2
c = 3
For a shift to the left by π/2 gives, we have,
d = π/2
Thus, the desired equation of the drawn cosine graph is,
y = 5 cos [( π/40)(x - (π/2))] + 3
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Related Questions
Curves defined using parametric equations have an orientation
true or false?
Answers
Answer:
True
Step-by-step explanation:
So this problem is asking whether Parametric because have an orientation and the short answer to that is that this is true. Parametric curves do have an orientation and we'll just do a little demonstration to share. Why? So let's take these very standard privatization of the unit circle where X is he going to cause of tea and why is equal to sign of tea now? The final part of defining the characterization is defining that the main range of tea on will take t to go from zero to two pi Andi, In this case, this is the this parts of circle going in this direction in the counter clockwise direction. And this is because as X as he goes from zero to play over two x decreases and signed increases on it sets us off on this direction as if that's one orientation off this curve of the circle. But we can alter this curve to get a different orientation. So if we were to take the Parametric equation given by X is equal to the co sign a minus T and why is equal to the sign of minus T in this case? Well again, take the domain to be from 0 to 2 pi. But because of the mind, the sign when it the mind sign doesn't actually affect X, it doesn't effect for co sign Well, it does affect the sign of minus T. So as T increases, the X value still decreases. But the Y value initially decreases as well, so this sets the arrows off on the clockwise direction. So this is another orientation off the same curve. So we've got two very different characterizations that describe the same physical curve. But because of the differences, the curve is constructed in opposite direction, and that is what the orientation defines. So it's very important to make sure that when you're drawing and describing the curve, you're parametric equations are in the correct orientation because otherwise you may encounter some difficulties later on. Because you'll be working with equation start that are, in fact, wrong
Thank you,
Eddie.
Given f (x) = 4x + 3 – 2, what is f –1(x)?
f –1(x) = log4(x – 3) + 2
f –1(x) = log4(x + 3) – 2
f –1(x) = log4(x – 2) + 3
f –1(x) = log4(x + 2) – 3
Answers
The inverse function of f(x) = 4^(x + 3) -2 is f-1(x) = log₄(x + 2) - 3
How to determine the inverse function?The function is given as:
f(x) = 4^(x + 3) -2
Rewrite as:
y = 4^(x + 3) -2
Swap x and y
x = 4^(y + 3) -2
Add 2 to both sides
x + 2 = 4^(y + 3) -2 + 2
This gives
x + 2 = 4^(y + 3)
Take the logarithm of both sides
log(x + 2)= log(4)^(y + 3)
Apply the base rule of logarithm
log(x + 2)= (y + 3)log(4)
Divide both sides by log(4)
log(x + 2)/log(4) = y + 3
Apply the change of base rule of logarithm
log₄(x + 2) = y + 3
Subtract 3 from both sides
log₄(x + 2) - 3 = y + 3 - 3
Evaluate the difference
log₄(x + 2) - 3 = y
Rewrite as:
y = log₄(x + 2) - 3
Rewrite as:
f-1(x) = log₄(x + 2) - 3
Hence, the inverse function of f(x) = 4^(x + 3) -2 is f-1(x) = log₄(x + 2) - 3
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The hospital in patient admission rate for employees of the ramsey
Answers
The computation shows that the the hospital in-patient admission rate for the employees of the Prendergast Corporation is 104.5 admissions per 1000 employees.
How to calculate the value?The in-patient admission rate for Ramsey Manufacturing company = 100 admissions per 1000 employees.
It is given that the in-patient rate of Prendergast will be 10% higher than Ramsey's.
So, in-patient rate of Prendergast:
= 100 + 10% of 100
= 110 admissions
Now, it is also given that out of total in-patients, 5% are for tonsillectomies (which are not in-patient).
So, in-patient admission rate for Prendergast:
= 110 - 5% of 110
= 110 - 5.5
= 104.5 admissions per 1000 employees
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Complete question:
The hospital in-patient admission rate for employees of the Ramsey Manufacturing Company is 100 admissions per 1000 employees per year.
The Prendergast Corporation has older employees and is told that its hospital in-patient admission rate will be 10% higher than Ramsey’s. At the same time, however, all tonsillectomies will be done in the hospital’s outpatient department instead of requiring the patient to be admitted as an in-patient. Given that 5% of all in-patient admissions are for tonsillectomies, what is the hospital in-patient admission rate for the employees of the Prendergast Corporation?
A. 110.0 B. 115.0 C. 109.5 D. 100.0 E. 104.5 F. 117.0
A certain radioactive material decays in such a way that the mass in kilograms remaining after t years is given by the function
m(t)=120e−0.018t
How much mass remains after 50 years?
Answers
The mass of radioactive material remaining after 50 years would be 48.79 kilograms
How to determine the amountIt is important to note that half - life is the time it takes for the amount of a substance to reduce by half its original size.
Given the radioactive decay formula as
m(t)=120e−0.018t
Where
t= 50 years
m(t) is the remaining amount
Substitute the value of t
[tex]m(t) = 120e^-^0^.^0^1^8^(^5^0^)[/tex]
[tex]m(t) = 120e^-^0^.^9[/tex]
Find the exponential value
m(t) = 48.788399
m(t) = 48.79 kilograms to 2 decimal places
Thus, the mass of radioactive material remaining after 50 years would be 48.79 kilograms
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IM IN A HURRY PLEASE HELP ME QUESTION IS DOWN BELOW WORTH 15 POINTS each
Answers
If the radius of the circle is 5 m and angle subtended by arc LKM is 70° then the length of the arc LNM is 25.29 m.
Given that the radius of the circle is 5 m and angle subtended by arc LKM is 70°.
From the figure given we can estimate that the length of arc LNM can be circumference of circle subtracted by length of arc LKM.
Circulference of the circle is basically the length of whole arc of the circle. It is also know as perimeter of the circle. The formula of calculating circumference of a circle is 2πr in which r is the radius.
So,
length of arc LNM=Circumference of circle-length of arc LKM.
We know that length of an arc =Θr
in which Θ is the angle in radian.
length of arc=70*π[tex]/180*5[/tex]
=70π[tex]/36[/tex]
Circumference of circle=2π*5
=10π
Length of arc LNM=10π-70π[tex]/36[/tex]
=(360π-70π)[tex]/36[/tex]
=290π[tex]/36[/tex]
Put π=3.14
=290*3.14[tex]/36[/tex]
=910.6[tex]/36[/tex]
=25.29m
Hence the length of arc LNM is 25.29 m.
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if x was 4, what is the value of 4x + 3
Answers
Answer:
19
Step-by-step explanation:
4(4)+3
16+3=19
Answer:
19
Step-by-step explanation:
4x+3
4(4)+3
16+3
19
hope it helps
A rain gutter is to be made of aluminum sheets that are 16 inches wide by turning up the edges 90. See the illustration.
(a) What depth will provide maximum cross-sectional area and hence allow the most water to flow?
(b) What depths will allow at least 30 square inches of water to flow?
Answers
Based on the width of the aluminum sheets and the angle of the edges, the depth that allows maximum cross-sectional area is 4 inches.
How much can the maximum cross-sectional area?Assuming that the depth is x, the area would be:
= x (16 - 2x)
= 16x - 2x²
The maximum of the parabola is:
x = -b / 2a
= - 16 / (2 x -2)
= -16 / -4
= 4 inches
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To calculate the area of this shape, you would identify the known figures and add their individual areas together.
However, the half circle is unlike the other known figures in this unknown shape in that it TAKES AWAY from the total area. How should you handle its area when
calculating the total area of the unknown figure?
Answers
When calculating the total area of this unknown figure, you should use (π - r²)/4 for the area of both the semicircle (half circle) and the larger square.
How to determine the total area?By critically observing this unknown figure, we can logically deduce that it comprises a semicircle, squares, a rectangle and a triangle. Thus, its total area should be calculated by adding the area of each of the geometric figures together.
Total area = A₁ + A₂ + A₃ + A₄
However, you should take note that the diameter of the semicircle (half circle) is equal to the side length of the larger square.
Area of a semicircle = πr²/4
Thus, the total area of both the semicircle (half circle) and the larger square is given by:
Area = πr²/4 - d²
Note: d = 2r
Area = πr²/4 - (2r)²
Area = πr²/4 - 4r²
Area = (π - r²)/4 sq. units.
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A company's year-end data shows the following amounts. Current assets $65,000 Current liabilities $25,000 Net income $7,500 Net sales $125,000 Total assets $150,000 Based on this information, what is the company's current ratio?
Answers
The company's current ratio is 2.6.
What is the current ratio?
Current ratio is an example of a liquidity ratio. Liquidity ratios are financial ratios measure a firm's ability to honour its short terms obligations.
Current ratio = current asset /current liability
$65,000 / 25,000 = 2.6
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Which of the following is the most accurate statement about Spatial Relations?
A- Children understand the spatial world watching movies about animals all over the world.
B- Children Need to know, use and be able to read spatial terms before they can demonstrate spatial awareness.
C-Spatial terms such as above, below, behind, and under are used to describe objects in space from different perspectives and are important for children to understand and express as they understand spatial relations.
Answers
Option C. The correct statement about spatial relationshipo is that Spatial terms such as above, below, behind, and under are used to describe objects in space.
What are spatial relationship?These are the types of relationships that are used to relate objects to the location of somethings else. It makes use of words like under, above etc.
It is one of the knowledge that children need in the preoperational stage to learn how to use logical operations.
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I can't find the answer for 46.3 divided by 1.5 Help!
Answers
Is that what your looking for??
The solution to the problem when [tex]46.3[/tex] divided by [tex]1.5[/tex] the result of the division is [tex]30.8666667[/tex] in decimal form.
When a large number is distributed into small numbers of equal parts is called division.
The number [tex]1.5[/tex] is not a factor of [tex]46.3[/tex] , on dividing them the remainder is not zero.
Therefore, first convert [tex]1.5[/tex] it into a whole number by multiplying it [tex]10[/tex] so it becomes [tex]15[/tex] .
Then convert [tex]46.3[/tex] it into a whole number by multiplying it by [tex]10[/tex] so it becomes [tex]463[/tex]
Here, [tex]\dfrac{463 \times 10}{15 \times 10}[/tex] which results in [tex]\dfrac{463}{15}[/tex].
Now, divide [tex]\dfrac{463}{15}[/tex] , the result will be [tex]30.8666667[/tex].
The solution to the problem when [tex]46.3[/tex] divided by [tex]1.5[/tex] the result is [tex]30.8666667[/tex].
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The day you were born your Grandparents started to save money for your future education. They decided to deposit $3,000 at the end of each year, for eighteen years, in a savings account that pays 7.5% per year, compounded annually.
On your 19th birthday you get admission into university for a four-year degree course. You calculate that you require approximately $35,000, for your fees, books and living expenses for each year. Assuming you keep the money in the bank accruing the same interest, and withdraw $ 35,000, at the beginning of each year.
a) Calculate the total value of this investment at the end of the term?
b) Determine the total interest earned?
Will your inheritance last you for the full tenure of your four years at the university?
Answers
The total value of the investment is $107,032.16.
The total interest earned is $53,032.16
The inheritance would last the tenure of the university.
What is the total value of the investment?The formula that can be used to determine the future value of the 18-year annuity is: annuity factor x amount deposited yearly
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest rate n = number of years$3000 x [(1.075^18 - 1) / 0.075] = $107,032.16
Amount deposited in the course of 18 years = (3000 x 18) = 54,000
Interest earned = 107,032.16 - 54,000 = $53,032.16
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What is the standard form equation of an ellipse that has vertices (−2,14) and (−2,−12) and co-vertices (9,1) and (−13,1)?
Answers
Answer:
[tex]\frac{(x+2)^2}{121} + \frac{(y-1)^2}{169} = 1[/tex]
Step-by-step explanation:
First, we identify that the vertices are vertical.
(an image below is attached if you want to see a visualization)
Therefore, the equation is such:
[tex]\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1[/tex]
(h,k) is the center of the ellipse, which we could easily calculate by finding the midpoint between any pair of vertices.
This gets us (-2, 1)
Therefore, h = -2, k = 1
Now we want to find a,
we know the length of the major axis is 2a, and in this case, our length of the major axis is 26. So a = 13
Now we want to find b,
We know the length of the minor axis is 2b, and in this case, our length is 22, so b = 11
Now we will just plug in all the values and simplify to get our answer! B)
help please I really need it
Answers
the ammount of time between interest payments is know as
Answers
The amount of time between interest payments is know as period.
What is period?Period is the time between when interests are paid. Period can vary depending on the terms of the investment. Periods can be yearly, quarterly, semi-annually, biweekly or weekly.
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A retailer buys a table costing £95 and sells it for £114 what is the name percentage change
Answers
Answer:
20% increase
Step-by-step explanation:
To find the percent increase, use this formula
New value - original value
----------------------------------------
original value
114-95
-----------
95
19
---- = .2
95
.2 is 20%, so it is a20% increase
(SAT prep) Find the value of x in each case of the following exercises:
Answers
The value of x from the figure is 70°
How to determine the valueIt is important to note that alternate angles are equal and angles in a triangle sum up to 180°
We then have that,
x + 50 + 60 = 180 °
x + 110 = 180°
Make 'x' subject
x = 180 - 110
x = 70 °
Thus, the value of x from the figure is 70°
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A tennis ball can in the shape of a right circular cylinder holds six tennis balls snugly. If the radius of a tennis ball is 3.4 cm, what percentage of the can is not occupied by tennis balls?
Answers
The percentage of the can that is not occupied by tennis balls is 33.33%.
PercentageGiven:
Radius of cylinder=3.4cm
First step
Height of cylinder=3.4×2×6
Height of cylinder=40.8 cm
Second step
Volume of cylinder=(3.4)^2×40.8
Volume of cylinder=471.648
Volume of tennis=6×4/3×(3.4)^3
Volume of tennis=314.432
Third step
Percentage=471.648-314.432/471.648×100
Percentage=33.33%
Therefore the percentage of the can that is not occupied by tennis balls is 33.33%.
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In London today, twice the high temperature was more than five times the high temperature minus eighty-seven. In interval form, what are the possible temperatures?
Answers
The possible temperatures are represented by (-∝, 29)
How to determine the possible temperatures?Let the high temperature be x
So, the statement can be represented as:
2x > 5x - 87
Subtract 5x from both sides
-3x > -87
Divide both sides by -3
x < 29
In interval form, we have (-∝, 29)
Hence, the possible temperatures are represented by (-∝, 29)
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A rock is dropped from a height of 100 feet. Calculate the time between when the rock was dropped and when it landed. If we choose "down" as positive and ignore air friction, the function is h (t) = 36t2 - 64
t= 3.56 seconds
t= 1.78 seconds
t= 1.33 seconds
t= 8 seconds
Answers
The time it takes between when the rock was dropped and when it landed is 1.33 seconds.
What is the time it takes for the rock to land?The time it takes the rock to land on the ground from where it is dropped can be determined by using the function given.
The height of the function when it hits the ground will be zero, so the function can be written as:
[tex]\mathbf{h(t) =36t^2 -64}[/tex]
[tex]\mathbf{0 =36t^2 -64}[/tex]
64 = 36t²
[tex]\mathbf{t^2 = \dfrac{64}{36}}[/tex]
t² =1.778
[tex]t = \sqrt{1.778}[/tex]
t = 1.333 seconds
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Alex, Barry, Charles, David, and Eric paint houses for a living. Alone, Alex can paint an entire house in 24 hours, Barry in 26 hours, Charles in 20 hours, David in 21 hours and Eric in 27 hours. If they work together, approximately how long will it take them to paint 3 houses, rounded to the nearest hour?
Answers
It would take 14 hours for all of them to paint the three houses.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let t represent the time it would take them working together, hence:
(1/24 + 1/26 + 1/20 + 1/21 + 1/27)t = 3
t = 14 hours
It would take 14 hours for all of them to paint the three houses.
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what’s the equation of this line?
Answers
Answer:
Step-by-step explanation:
Hello : let A(0,-6) B(8,-8)
the slope is : (YB - YA)/(XB -XA)
(-8+6)/(8-0) = -2/8=-1/4
an equation is : y=ax+b a is a slope
y = (-1/4)x +b
the line through point A (0;-6) : -6 = (-1/4)(0)+b
b = -6 the equation is : y =(-1/)4x-6
Identify the coordinates of the vertices after the transformation
Answers
Answer:
A' is (-2,-1)
B' is (3,0)
C' is (2,4)
Step-by-step explanation:
Find u. v.
U V=
(8, 2)
= (-6, 3)
U =
V =
Answers
Answer:
-42
Step-by-step explanation:
Assuming you want the dot product,
(8)(-6)+(2)(3)= -42
What is the measure of ABC in the figure below?
Answers
Since Angle ABD and Angle DBC are complementary angles, the measure of Angle ABC is 90 degrees.
Hence, option D is the correct answer.
What is the measure of ABC in the figure below?Two angles are called complementary when their measures add to 90 degrees.
From the diagram; Angle ABD and Angle DBC are complementary angles that add up to make 90 degrees.
Since Angle ABD and Angle DBC are complementary angles, the measure of Angle ABC is 90 degrees.
Hence, option D is the correct answer.
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15. Solve the following inequality: 38 < 4x + 3 + 7 - 3x.
OA.x < 4
OB.x>4
O C. x < 28
OD. x > 28
Answers
Step-by-step explanation:
38<4x+3+7-3x
4x+3+7>38
4x+3+7<3x
X>7
X<-10
= Ø
3. Consider the equation 2x² + 4y² + 8x - 16y + F = 0. Find all values of F such that the graph of the equation is a) an ellipse b) a point - ellipse
Answers
The equation 2•x² + 4•y² + 8•x - 16•y + F = 0, at a value of F is the graph of an ellipse when we have;
a. A value of F that results in the equation of an ellipse is; F = 23
b. A point on the ellipse is ((-2), 2.5)
Which method can be used to find the value of F that gives the graph of an ellipse?The equation of an ellipse can be presented as follows;
[tex] \frac{(x - h) ^{2} }{ {a}^{2} } + \frac{(y - k) ^{2} }{ {b}^{2} } = 1[/tex]
The given equation can be presented as follows;
2•x² + 4•y² + 8•x - 16•y + F = 0
Collecting like terms, gives;
2•(x² + 4•x) + 4•y² - 16•y + F = 0
Adding a constant value to both sides, we get;
2•(x² + 4•x + 4) + 4•y² - 16•y + 16 + F = 24
2•(x + 2)•(x + 2) + (2•y - 4)•(2•y - 4) + F = 24
2•(x + 2)² + 4•(y - 2)² + F = 24
2•(x + 2)² + 4•(y - 2)² + F - 23 = 24 - 23 = 1
When F = 23, we have;
2•(x + 2)² + 4•(y - 2)² + 23 - 23 = 1
2•(x + 2)² + 4•(y - 2)² = 1Which gives;
[tex] \mathbf{\frac{(x + 2) ^{2} }{ \left( {\frac{1}{ \sqrt{2} }} \right)^{2}} + \frac{(y - 2) ^{2} }{ \left( {\frac{1}{ 2} } \right)^{2}} } = 1[/tex]
Where;
a = 1/(√2)b = 1/2b) A point on the ellipse can be found as follows;
2•(x + 2)² + 4•(y - 2)² = 1When x = -2
2•((-2) + 2)² + 4•(y - 2)² = 1
0 + 4•(y - 2)² = 1
y - 2 = √(1/4) = 1/2
y = 2 + 1/2 = 2.5A point on the ellipse is therefore;
((-2) , 2.5)When y = 2
2•(x + 2)² + 4•(2 - 2)² = 1
2•(x + 2)² + 0 = 1
2•(x + 2)² = 1
(x + 2)² = 1/2
x + 2 = √(1/2)
x = √(1/2) - 2
Another point on the ellipse is therefore;
((√(1/2) - 2), 0)
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a right triangle has side lengths of 6, 10, and x. find x if it is the hypotenuse
Answers
X= root 6^2 + 10^2
X=2 root 34
So, x is 11.66 to 2d.p
(root means square root)
Hope this helps!
Answer:
x = 11.66
Step-by-step explanation:
Hello!
We can use the Pythagorean theorem to solve for the hypotenuse.
Pythagorean Theorem: [tex]a^2 + b^2 = c^2[/tex]
a = legb = legc = hypotenusePlug the values of 6 and 10 into the formula to solve.
Find the Hypotenuse[tex]a^2 + b^2 = c^2[/tex][tex]6^2 + 10^2 = c^2[/tex][tex]36 +100 = c^2[/tex][tex]136 = c^2[/tex][tex]c = \sqrt{136[/tex][tex]c \approx 11.66[/tex]The hypotenuse is approximately 11.66.
Why is it important for the world to have an order of operations
Answers
Answer:
Well a sovial reason is that no one would no what order yo do it in, which would cause arguments
Answer:
without the order of operations, maths literally doesn't work.
Step-by-step explanation:
like whilst some equations don't rely on a particular order for solving, if you for example, don't solve brackets first, then try to multiply them by something else basically the equation breaks and you get the wrong answer. So... kids remember your BODMAS :)
solve the inequaility - x/4 <2
Answers
❄ Hi there,
let us start by considering this: What should we do to get x all by itself?
Since it's
divided by 4multiplied by -1we should
multiply (both sides) by 4divide (both sides) by -1So, let's start.
[tex]\sf{\dfrac{-x}{4}\times4 < 2\times4}}[/tex]
[tex]\sf{-x < -8}[/tex]
[tex]\sf{x > 8}[/tex]
When we divide both sides of an inequality by a negative number, we do "the inequality flip" and flip the inequality sign.
❄