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cept and each
maximum or minimum point. List the period and amplitude.

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2.I am trying to figure out the optimal radius that will give the lowest surface area of a cylinder. I ...

have done the calculus which reveals that the surface area is at a minimum when height is double the radius. I am now trying to find an equation for the relationship between the amount of wasted surface area as a percentage of the minimum surface area and the ratio between height and radius.
If I were to plot it on a graph, the y axis would be the percentage of excess materials needed as a percentage of the minimum possible surface area, and the x axis would be height divided by radius. Since the surface area is minimized when height=2(radius), I know that when x=2, y=0.
The website https://www.datagenetics.com/blog/august12014/index.html explains what I am trying to do quite well and shows the graph below. I am trying to find the equation for this graph, but am unsure how to go about it.

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hin a free demo session as I have my answers, but just want to confirm them, that would be greatly appreciated.
Question 1:
A block of mass M = 0.10 kg is attached to one end of a spring with spring constant k = 100 N/m . The other end of the spring is attached to a fixed wall. The block is pushed against the spring, compressing it a distance x = 0.04 m . The block is then released from rest, and the block-spring system travels along a horizontal, rough track. Data collected from a motion detector are used to create a graph of the kinetic energy K and spring potential energy Us of the system as a function of the block's position as the spring expands. How can the student determine the amount of mechanical energy dissipated by friction as the spring expanded to its natural spring length?
Question 2:
The Atwood’s machine shown consists of two blocks connected by a light string that passes over a pulley of negligible mass and negligible friction. The blocks are released from rest, and m2 is greater than m1. Assume that the reference line of zero gravitational potential energy is the floor. Which of the following best represents the total gravitational potential energy U and total kinetic energy K of the block-block-Earth system as a function of the height h of block m1?
Question 3:
A 2 kg block is placed at the top of an incline and released from rest near Earth’s surface and unknown distance H above the ground. The angle θ between the ground and the incline is also unknown. Frictional forces between the block and the incline are considered to be negligible. The block eventually slides to the bottom of the incline after 0.75 s. The block’s velocity v as a function of time t is shown in the graph starting from the instant it is released. How could a student use the graph to determine the total energy of the block-Earth system?
Question 4:
A block slides across a flat, horizontal surface to the right. For each choice, the arrows represent velocity vectors of the block at successive intervals of time. Which of the following diagrams represents the situation in which the block loses kinetic energy?

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a jump discontinuity. You then need to write and find the limit at that point of discontinuity and jump discontinuity. You also have to find the limit at a point on your piecewise defined function that is continuous. Each limit needs to be justified using the graph and the limit notation. Your piecewise defined function needs to have a beginning and end. That means that the domain cannot go to infinity. You will state the domain and the range using interval notation.

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hoots a laser beam in a straight path from a deck 14 feet above sea level off the coast of Florida, is it possible the laser beam will cross the path of the model rocket? Use a piece of graph paper to write/model equations f ( x ) to represent the path that models the rocket and g ( x ) to represents the path that models the laser beam.

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function
C = 65t + 30
where t is the time in months.
(a) Calculate the gym membership cost over a six month period.
(b) Sketch the graph of the function C = 65t + 30 for t ≥ 0.
(c) Calculate the time, t, in months, when the total cost reaches 290 AUD.
In the neighbouring Nicolo’s Gym, the initial payment is 75 AUD higher than in
Paolo’s Gym, however the monthly fee is lower at 30 AUD per month.
(d) Determine the number of months it takes for the total cost to be less by
attending Nicolo’s Gym in comparison to Paolo’s Gym.

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itially, the balloon contains 4 ft3 of hydrogen and it takes 14 seconds to com-
pletely fill the balloon.
(a) At what rate is the balloon filled?
(b) Find a linear function V that models the volume of hydrogen in the balloon at
any time t.
(c) What can you say about the graph of V ? (You do not need to sketch the graph.)

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u decide that you should sell at least thirty items, but do not want to exceed 120 items. Based on a small survey of students, you also decide that the number of T-shirts should be at least twice the number of sweatshirts.
A. Assign variables to the unknown quantities and write a system of inequalities that model the given restrictions.
B. Graph the system, indicating an appropriate window and scale and shading the feasible region.
C. Determine the vertices of the polygonal feasible region.
D. Assume the profit on each sweatshirt is $5 and the profit on each T-shirt is $2. What is the maximum profit you can obtain?
E. How many sweatshirts and how many T-shirts should you sell to maximize your profit?

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rts to the fireworks platforms: one part is on the ground and the
other part is on top of a building. You are going to graph all of your results on one
coordinate plane. Make sure to label each graph with its equation. Use the following
equations to assist with this assignment.
• The function for objects dropped from a height where t is the time in
seconds, h is the height in feet at time it t, and 0 h is the initial height is
2
0 ht t h ( ) 16 =− + .
• The function for objects that are launched where t is the time in seconds, h is
the height in feet at time t, 0 h is the initial height, and 0 v is the initial velocity
in feet per second is 2
0 0 ht t vt h ( ) 16 =− + + .
Select the link below to access centimeter grid paper for your portfolio.
Centimeter Grid Paper
Task 1
First, conduct some research to help you with later portions of this portfolio
assessment.
• Find a local building and estimate its height. How tall do you think the
building is?
• Use the Internet to find some initial velocities for different types of fireworks.
What are some of the initial velocities that you found?
Task 2
Respond to the following items.
1. While setting up a fireworks display, you have a tool at the top of the
building and need to drop it to a coworker below.
a. How long will it take the tool to fall to the ground? (Hint: use the first
equation that you were given above, 2
0 ht t h ( ) 16 =− + . For the building’s
height, use the height of the building that you estimated in Task 1.)
b. Draw a graph that represents the path of this tool falling to the
ground. Be sure to label your axes with a title and a scale. Your graph
should show the height of the tool, h, after t seconds have passed.
Label this line “Tool”.

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ing 9.8 m/s after 1 second, 19.6 m/s after 2 seconds, and so on. Use the data values in your table to sketch a rough graph of velocity versus time for the 10° angle and another for the 40° angle. What value do the slopes of these graphs represent? Which graph has the greater slope? Why?

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12.A candle 6 inches high burns at a rate of 1 in every 2 hours for 5 hours. Write a function ...

els the given situation. Draw a graph and name the domain and range.

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risen to 28. Create the exponential formula for this
situation. Use appropriate variables other than ‘x’ and ‘y’ in your formula.
Create the graph of this formula in Excel and label the axes properly. Display
your equation on the graph.

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14.A quadratic relation has an equation of the form y = a(x - r) * (x - s) . The ...

has zeros at (2, 0) and (- 6, 0) and passes through the point (3, 5) . Determine the value of a.

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15.The graph for Airplane A shows the speed at which it travels as a function of time. The graph for ...

rplane B shows the distance it travels as a function of time.
Use the drop-down menus to complete the statements below about the two airplanes.

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lve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. Your work must be expressed in standard mathematical notation rather than calculator syntax.
Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
Let f be a twice-differentiable function such that f′(2)=0 . The second derivative of f is given by f′′(x)=x2e2−x−1 for 0≤x≤6 .
(a) On what open intervals contained in 0
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ed and an altitude of 35,000 feet. In fact, the distance the airplane travels at cruising speed is directly proportional to the time it travels. Using complete sentences, describe what the points (0, 0) and (4, 2268) represent.

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nbow is the shape of a parabola.
The equation for this parabola is y = -x2 + 36.
Graph of a parabola opening down at the vertex 0 comma 36 crossing the x–axis at negative 6 comma 0 and 6 comma 0.
Create a table of values for a linear function. A drone is in the distance, flying upward in a straight line. It intersects the rainbow at two points. Choose the points where your drone intersects the parabola and create a table of at least four values for the function. Remember to include the two points of intersection in your table.
Analyze the two functions. Answer the following reflection questions in complete sentences.
What is the domain and range of the rainbow? Explain what the domain and range represent. Do all of the values make sense in this situation? Why or why not?
What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.
Is the linear function you created positive or negative? Explain.
What are the solutions or solution to the system of equations created? Explain what it or they represent.

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? Explain your reasoning.
a) Postage rates per ounce
b) Charges at a parking lot per hour
c) Taxicab fares determined by the distance traveled
d) Temperatures over the course of an afternoon

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r the graph in
point-slope form. Explain your
steps.
3. Write the equation for the graph in
slope-intercept form. Explain your
steps.
4. Which equation is the easiest to
write by looking at the graph?
Explain why your choice is easier
than the other two equations.

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r the graph in
point-slope form. Explain your
step
3. Write the equation for the graph in
slope-intercept form. Explain your
steps.
4. Which equation is the easiest to
write by looking at the graph?
Explain why your choice is easier
than the other two equations.

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22.I don't understand how to graph a sin or cos equation by just looking at it. Vice versa, I have ...

dea how to write an equation looking at the graph.

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23.At which point on the graph of f(x) = -x(x+3)(x-3)^2 is the instantaneous rate of change equal to 0? ...

a) (-3,0) b) (0,0) c) (1, -16) d) (3,0)

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24.For which value(s) of a does the curve y = x^2 + ax + 2 ...

1 ? (Without using differentiation rules)
For each statement, explain why it must be true, or use an example to show that it can be false.
a)If y = f ( x ) has a horizontal tangent line at x = 1 then y = g ( x ) , where g ( x ) = f ( x − 1 ) + 1 , has a horizontal tangent line at x = 2 .
b)A tangent line always has exactly one point in common with the graph of the function.

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25.I dont understad these questions: For which value(s) of a does the curve y = x^2 + ...

ave a horizontal tangent line at x = 1 ? (Without using differentiation rules)
For each statement, explain why it must be true, or use an example to show that it can be false.
a)If y = f ( x ) has a horizontal tangent line at x = 1 then y = g ( x ) , where g ( x ) = f ( x − 1 ) + 1 , has a horizontal tangent line at x = 2 .
b)A tangent line always has exactly one point in common with the graph of the function.

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26. Value: 1 equation image indicator a. (x - 2)2(x - 3)2 b. (x2+ 4)(x2+ 9) c. (x - 2)(x + ...

2)(x - 3)(x + 3)
d. (x2 - 4)(x2+ 9)
Value: 1
The table below shows the cost of purchasing a standard stapler at five office supply stores, A through E. If the median cost of purchasing a standard stapler for these stores was $17.99, which of the following could NOT have been the cost of the stapler for Store A?
staplergraph.jpg
a. $19.95
b. $18.95
c. $16.95
d. $19.25
Value: 1
If equation image indicator then x =
a. 7
b. 1/5
c. 5
d. 1/7
Value: 1
A six−sided die, with sides numbered 1,2, 3,4,5, and 6, is tossed. What is the probability of tossing a number less than three?
a. 1/3
b. 0
c. 1/2
d. 1/4
Value: 1
If 6m + 4 = 8m, then 4m =
a. 6
b. 2
c. 8
d. 4
Value: 1
In the xy-plane, what is the y-intercept of the graph of the equation equation image indicator?
a. 2
b. 4
c. 16
d. There is no y-intercept.
Value: 1
Which of the following equations has both 2 and −4 as solutions?
a. x2 + 6x + 8 = 0
b. x2 - 2x - 8 = 0
c. x2 + 2x - 8 = 0
d. x2 - 2x + 8 = 0
Value: 1
The perimeter of a square is 20 ft. If you increase the length of the square by 2 feet and decrease the width by 1 foot, what is the area, in square feet, of the new figure?
a. 22
b. 28
c. 35
d. 40
Value: 1
(3x-2y4)-3 =
a. equation image indicator
b. equation image indicator
c. equation image indicator
d. equation image indicator
Value: 1
A softball is tossed into the air upward from a first floor balcony. The distance of the ball above the ground at any time is given by the function, distance function.png, where h(t) is the height of the softball above the ground (in feet) and t is the time (in seconds). What was the maximum height, in feet, of the softball above the ground after it was thrown?
a. 28
b. 30
c. 32
d. 34
Value: 1
A group of 100 people, some students and some faculty, attended a museum opening. Each student paid $10 per person for entrance to the museum and each of the faculty paid $25 per person for entrance. If the total paid, for all 100 people, was $1300, how many students attended the museum opening?
a. 20
b. 50
c. 70
d. 80
Value: 1
The ratio of Sam's age to Hank's age is 5 to 3. If the sum of their ages is 24, how old is Hank?
a. 21
b. 15
c. 19
d. 9
Value: 1
In the xy−coordinate plane shown below, point P has coordinates (8, −6). Which of the following is an equation of the line that contains points O and P?
O and P graph.jpg
a. equation image indicator
b. equation image indicator
c. equation image indicator
d. equation image indicator
Value: 1
The variables x and y are inversely proportional, and y = 2 when x = 3. What is the value of y when x = 9?
a. 54
b. 6
c. 2/3
d. 3/2
Value: 1
A farmer has 1235 trees to be planted on a rectangular parcel of land. If there are 24 trees planted in each row and each row must be complete before it is planted, how many trees will be left over after planting?
a. 21
b. 11
c. 0
d. 55

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that models profit earned is D = n(54 – n) – 10n. I need to find the vertex of this equation, and what does the vertex tell me about this situation.. For what x-values is the function increasing? Decreasing? What does this mean in terms of daily profit for Water World? Rewrite the function in vertex form. . Solve the equation 0 = n(54 – n) – 10n for n. Describe your solution method. How are the solutions from part (e) related to the graph of this function? Are the solutions real or complex? How do you know? What do the solutions from part (e) tell you about this situation?

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1.AU MAT 120 Systems of Linear Equations and Inequalities Discussion

mathematicsalgebra Physics