Answer:
VII
Step-by-step explanation:
You are on a team of architects. You are charged with building a scale-model replica of one section of a new roller coaster before construction gets underway.
Certain reinforcement cables and struts are required to make the roller coaster sturdier. The goal for this project is for your team to determine where to place these cables or struts. The mathematical models for these reinforcements are known.
Your team must provide both algebraic and graphical evidence for your conclusions regarding the location of the cables.
Directions:
The shape of this particular section of the rollercoaster is a half of a circle. Center the circle at the origin and assume the highest point on this leg of the roller coaster is 30 feet above the ground.
1. Write the equation that models the height of the roller coaster.
The circle equation is x² + y² = r²
Start by writing the equation of the circle. (Recall that the general form of a circle with the center at the origin is x2 + y2 = r2. (10 points)
The roller coaster's leg is 30 feet above the ground, as stated.
⇒ r = 30
⇒x² + y² = 30²
⇒x² + y² = 900
Now solve this equation for y. Remember the roller coaster is above ground, so you are only interested in the positive root. (10 points)
Roller coaster height equation: - x2 + y2 = 900
As we know, x² + y² = 900
⇒y² = 900 - x²
⇒y = √900 - x²
2. Graph the model of the roller coaster using the graphing calculator. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
Model 1: One plan to secure the roller coaster is to use a chain fastened to two beams equidistant from the axis of symmetry of the roller coaster, as shown in the graph below:
You need to determine where to place the beams so that the chains are fastened to the rollercoaster at a height of 25 feet.
3. Write the equation you would need to solve to find the horizontal distance each beam is from the origin. (10 points)
To find the horizontal distance, we use the equation: x = 30^2-25^2
4. Algebraically solve the equation you found in step 3. Round your answer to the nearest hundredth. (10 points)
The horizontal distance is 16.58 feet
How to determine the equation:
The height is given as: 25 feet.
The roller coaster's length is 30 feet.
Use h to represent horizontal distance.
Given that the roller coaster represents the hypotenuse side length, the following equation can be used to calculate h
In step 3, we have: 30^2-25^2
This gives
Take the roots of both sides to get our answer
5. Explain where to place the two beams. (10 points)
The beam should be placed 8 feet from the center.
The struts are y = √(x + 8) and y = √(x − 4).
The struts are 2 feet apart at the location of the beam: √(x + 8) − √(x − 4) = 2
Solving:
√(x + 8) = 2 + √(x − 4)
x + 8 = 4 + 4√(x − 4) + x − 4
8 = 4√(x − 4)
2 = √(x − 4)
x − 4 = 4
x = 8
Model 2: Another plan to secure the roller coaster involves using a cable and strut. Using the center of the half-circle as the origin, the concrete strut can be modeled by the equation and the mathematical model for the cable is. The cable and the strut will intersect.
6. Graph the cable and the strut on the model of the roller coaster using the graphing calculator. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
7. Algebraically find the point where the cable and the strut intersect. Interpret your answer. (10 points)
root 2x+8=x-8
2x+8=x^2-16x+64
-x^2+18x-56=0
x^2-18x+56=0
x(x-4)-14(x-4)=0
(X-4)(x-14)=0
x=4 ,x=14
x=14
Model 3: Another plan to secure the roller coaster involves placing two concrete struts on either side of the center of the leg of the roller coaster to add reinforcement against southerly winds in the region. Again, using the center of the half-circle as the origin, the struts are modeled by the equations and. A vertical reinforcement beam will extend from one strut to the other when the two cables are 2 feet apart.
8. Graph the two struts on the model of the roller coaster. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
Recall that a reinforcement beam will extend from one strut to the other when the two struts are 2 feet apart.
9. Algebraically determine the x -value of where the beam should be placed. (15 points)
10. Explain where to place the beam. (10 points)
Step 1: Equation for Roller Coaster Height.
The general form for the equation of a circle at the origin is:
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Equation of a Circle (at origin):}}\\\\x^2+y^2=r^2\\\\\bullet \ \text{where 'r' is the radius of the circle}\end{array}\right}[/tex]
It is given that the height of the roller coaster is 30 feet. Thus, r = 30. Plug this into the equation above and solve for 'y.'
[tex]\Longrightarrow x^2+y^2=(30)^2\\\\\\\\\Longrightarrow x^2+y^2=900\\\\\\\\\therefore \boxed{\boxed{y=\sqrt{900-x^2} }}[/tex]
Step 2: Graph of Roller Coaster Model.
Refer to the attached image(s).
Step 3: Equation for Horizontal Distance of Beams.
Going back to the equation we got in step (1), we can solve for the variable ‘x.’
[tex]\Longrightarrow y=\sqrt{900-x^2}\\\\\\\\\therefore \boxed{\boxed{x=\pm\sqrt{900-y^2}}}[/tex]
Step 4: Algebraic Solution for Beam Placement.
Given a height, y = 25 ft. We can plug this into the equation above and solve for 'x.'
[tex]\Longrightarrow x=\pm\sqrt{900-(25)^2}\\\\\\\\\Longrightarrow x=\pm\sqrt{275}\\\\\\\\\Longrightarrow x=\pm5\sqrt{11}\\\\\\\\\therefore \boxed{\boxed{x\approx\pm16.58 \ ft}}[/tex]
Step 5: Placement of Beams.
The beams should be placed 16.58 feet from either side of the origin.
Step 6: Graph of Cable and Strut.
Refer to the attached image(s).
Step 7: Algebraic Intersection of Cable and Strut.
To find the point of intersection, set the two given equations (from model 2) equal to each other and solve for 'x.'
[tex]y=\sqrt{2x+8} \ \text{and} \ y=x-8\\ \\\\\\\Longrightarrow \sqrt{2x+8}=x-8 \\\\\\\\\Longrightarrow 2x+8=(x-8)^2 \\\\\\\\\Longrightarrow 2x+8=x^2-16x+64\\\\\\\\\Longrightarrow x^2-18x+56=0\\\\\\\\\Longrightarrow (x-14)(x-4)=0\\\\\\\\\therefore x=\{14,-4\}[/tex]
In order to determine which value of 'x' is correct we must verify the solution.
When x = 14:
[tex]\Longrightarrow 2(14)+8=(14-8)^2\\\\\\\\\Longrightarrow 36=36 \checkmark[/tex]
When x = -4:
[tex]\Longrightarrow 2(-4)+8=(-4-8)^2\\\\\\\\\Longrightarrow 0\neq 144[/tex]
Thus, the correct x-value is 14. Plug this into either given equations and solve for 'y.'
[tex]\Longrightarrow y =x-8\\\\\\\\\Longrightarrow y =14-8\\\\\\\\\therefore y=6[/tex]
Thus, the point of intersection is (14, 6).
Step 8: Graph of Two Struts.
Refer to the attached image(s).
Step 9: Algebraic Solution for Beam Placement.
To find the x-value of the beam placement, you need to find the point where the two strut equations are 2 feet apart. Set up an equation using the given functions and solve for 'x.'
[tex]\Longrightarrow y=\sqrt{x+8}-\sqrt{x-4}; \ y=2\\\\\\\\\Longrightarrow 2=\sqrt{x+8}-\sqrt{x-4}\\\\\\\\\Longrightarrow 2+\sqrt{x-4}=\sqrt{x+8}\\\\\\\\\Longrightarrow 4\sqrt{x-4}+x=x+8\\\\\\\\\Longrightarrow 4\sqrt{x-4}=8\\\\\\\\\Longrightarrow \sqrt{x-4}=2\\\\\\\\\Longrightarrow x-4=4\\\\\\\\\therefore \boxed{\boxed{x=8}}[/tex]
Step 10: Placement of Beam.
The beams will need to be placed 8 feet from either side of the origin.
Write an algebraic expression to represent the following situation: "Greg has nickels and pennies in his pocket. The number of pennies is 7 less than twice the number of nickels. Let N represent the number the nickels. Write an expression for the number of pennies."
Algebraic expression to represent the given situation would be, P = 2N - 7
What is algebraic expression?
When we utilize numbers and words to solve a mathematical problem, we are using algebraic expressions.
Let P represent the number of pennies in Greg's pocket.
We know that the number of pennies is 7 less than twice the number of nickels, which can be written as:
P = 2N - 7
This expression represents the number of pennies in terms of the number of nickels.
Therefore, Algebraic expression to represent the given situation would be, P = 2N - 7
To know more about algebraic expression visit,
https://brainly.com/question/4344214
#SPJ1
Special Triangle Segments......
3. Find the value of the variable given the triangles are similar. Answer the additional questions. Show all work. Name the property you used in your problem.
The value of x is 20 and the property is similar triangle
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.
Since ∆ ADC and IJH are similar, the ratio of their corresponding sides are equal.
Therefore,1.2/12 = 2/x
1.2x= 12×2
x = 24/1.2
x = 20
Therefore the value of x is 20
learn more about similar triangle from
https://brainly.com/question/14285697
#SPJ1
Help with this PLEASE!!
Answer:
x = 24°
Step-by-step explanation:
See the attached worksheet. Let a and b stand for the two unknown angales as shown on the worksheet. Since the sum of angles across a straight line is 180°, we can express the two angles as:
a = 180°-x, and
b = 180° - 111° or b = 69°
The sum of the interior angles of a quadrilateral is 360°. Set the four angles equal to 360° and solve for x.
x = 24°
What is the slope of a line perpendicular to a horizontal line?
The slope of a horizontal line perpendicular to a horizontal line is infinity or undefined.
The slope of a line is the that of the inclination that is makes with a horizontal line.
So, the slope of horizontal line is 1 because the value of tam(0 degrees) is also 1.
A line that is perpendicular to the horizontal line will have a the value of angle to be 90 degrees and we know that,
tan (90 degrees) = undefined/infinity
So, here we are concluding that the slope of any line that is being lying perpendicularly on any horizontally lying line is infinity or undefined.
To know more about slope, visit,
https://brainly.com/question/16949303
#SPJ4
I think I have the answer, but it keeps saying I'm incorrect.
Answer:
B.
D.
Step-by-step explanation:
B is actually 141, not 14.1
(204 plus 98) is more than 250.7
This means that multiplying it by 8.3 would not even make it near 250.7
It would actually be 2506.6
Answer: Your answers are B and D.
Solve 1/11 divided by 4
a
one forty fourth
b
44
c
four forty fourths
d
one fifteenth
Answer:
a ) one by forty fourth
Step-by-step explanation:
1/11 ÷ 4
= 1/11 x 1/4
= 1/44
therefore, the answer is one by forty fourth
Hope it's helpful..
Teacher's name Height with the heels Height without the heels
Ms. Mathey 190 cm 182.4 cm
Mrs. Meansy 162.5 cm 156 cm
Ms. Cool 140 cm 134.4 cm
Miss Petite cm 155.1 cm
Ms. Lora cm
159.8 cm
The required complete table of heights of teachers is shown in the image attached.
What is the percentage?The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
Here,
To calculate the height without heels, we need to subtract 4% of the height with heels from the height with heels. This can be done by multiplying the height with heels by 0.96 (which is equivalent to subtracting 4%):
Height without the heels = Height with the heels x 0.96
Using this formula, we can fill in the chart for each teacher.
Thus, the required complete table has been shown in the image attached.
Learn more about percentages here:
brainly.com/question/13450942
#SPJ1
how to convert milliliters to ounces?
From the given data by doing unit conversion, 250 mL is approximately equal to 8.4535 fluid ounces.
Unit conversion is the process of converting a given quantity from one unit of measurement to another.
To convert milliliters (mL) to fluid ounces (fl oz), you can use the following conversion factor:
1 mL = 0.033814 fl oz
To convert, simply multiply the number of milliliters by 0.033814 to get the equivalent volume in fluid ounces.
For example, if you want to convert 250 mL to fluid ounces, you can use the conversion factor as follows:
250 mL × 0.033814 fl oz/mL = 8.4535 fl oz (rounded to four decimal places)
Learn more about unit conversion here: brainly.com/question/97386
#SPJ4
a wheel for a wheel of winning game makes one revolution in 4 seconds. What is their unit rate in terms of minutes
It will take 15 revolutions per minute.
What is the unit rate?
An item's unit rate is its price for one of them. This is expressed as a ratio with a one as the denominator. For instance, if you covered 70 yards in 10 seconds, you covered 7 yards on average every second. Seven yards in one second and 70 yards in ten seconds are both ratios, but only one of them is a unit rate.
Here, we have
Given: a wheel for a wheel of winning game makes one revolution in 4 seconds.
Take a time for 1 revolution = 4 seconds
Assume that the unit rate in terms of revolutions per minute is x.
1 rev/4 second = x rev/60 second
4x = 60
x = 15
Hence, it will take 15 revolutions per minute.
To learn more about the unit rate from the given link
https://brainly.com/question/19493296
#SPJ1
What is the percent composition of water in the compound magnesium sulfate heptahydrate, MgDO4•7H2O?
A. 7.3%
B. 24.8%
C. 48.8%
D. 51.2%
Answer:51. 2%
Step-by-step explanation:
24+32+(16×4)+7(2+16)=24+32+64+126=246
26g of Epsom salt contains 126g of water of crystallisation.Hence, 100g of Epsom salt contains 100×126/246The % of H2O in MgSO4.7H2O=
Drag each tile to the correct box. Not all tiles will be used.
Order the steps for bisecting line segment AB using a reflective device.
A
Place the reflective device anywhere on
the segment.
Move the reflective device around on the
paper until the reflection of point A
coincides with point B.
Place the reflective device on Point A.
Place the reflective device on Point B.
Draw the line of reflection using the edge
of the reflective device as a guide.
The correct steps for bisecting line segment AB using a reflective device is given below:
Place the reflective device on Point A.Move the reflective device around on the paper until the reflection of point A coincides with point B.Draw the line of reflection using the edge of the reflective device as a guide.What is Bisecting a line?Cutting a line exactly in half is known as bisecting it. Due to the fact that the line you are drawing will be at a right angle to the initial line, it is also sometimes referred to as creating a perpendicular bisector.
A compass, pencil, and ruler are required.
Read more about bisecting a line here:
https://brainly.com/question/628636
#SPJ1
write a explicit formula in slope intercept form.
an=-3+5(n-1)
An explicit formula in slope intercept form for the given expression is aₙ = 5n - 8.
What is the slope-intercept form?Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.Based on the information provided above, we can logically deduce that an equation in slope-intercept form that models the situation is given by:
aₙ = -3 + 5(n - 1)
aₙ = -3 + 5n - 5
aₙ = 5n - 8
Read more on y-intercept here: brainly.com/question/6240745
#SPJ1
Need help understanding this question
Answer:
The number of large bars of chocolate sold is:
1/4 x 208 = 52
The number of small bars of chocolate sold is:
208 - 52 = 156
The total amount of money earned from selling the large bars of chocolate is:
52 x e1 = e52
The total amount of money earned from selling the small bars of chocolate is:
156 x 60p = £93.60
To convert e52 to pounds, we need to multiply by the exchange rate. Let's assume that the exchange rate is £1 = e1.20.
So, e52 = £52 x 1.20 = £62.40
Therefore, the total amount of money earned from selling all 208 bars of chocolate is: £62.40 + £93.60 = £156
Rafael runs 4 miles in 30 minutes. At the same rate, how many minutes would he take to run 10 miles?
Rafael will take 75 minutes to run 10 miles at the same rate.
Solution:
We know that Rafael can run 4 miles in 30 minutes.
So, we can find his pace, as follows:
30 minutes/ 4 miles = 7.5 minutes per mile.
So, for 10 miles he will take 7.5* 10 = 75 minutes.
Hence, it will take Rafael 75 minutes to run 10 miles.
Learn more about Unitary Method here:
https://brainly.com/question/28455000
Find the coordinates of the circumcenter of ABC with vertices A(-9, -4),
B(-9, 8), and C(-1,0).
The coordinates of the circumcenter are?
Answer:
Step-by-step explanation: 32 because 9 x4 lol
Sarah needs to earn a C in her Algebra class. Her current test scores are 69, 75, 65, and 83. Her final exam is worth 4 test scores. In
order to earn a C Sarah's average must lie between 70 and 79 inclusive. What range of scores can Sarah receive on the final exam to
earn a C in the course?
s final exam scores[
(Type an integer or a simplified fraction.)
Answer:
Sarah must get between a 67 and 85.
Step-by-step explanation:
69 + 75 + 65 + 83 = 292
70 ≥ 1/8(292 + 4x) ≥ 79
70 ≥ 0.5x + 36.5 ≥ 79
67 ≥ x ≥ 85
If 14 cars and 10 vans are in a parking lot what is the radio of cars to vans
Answer: 7:5
Step-by-step explanation:
Mr. Lee's house has two stories and an attic. The first floor is
9½ feet high, the second floor is 8 3/4 feet high, and the entire
house is 24 7/12 feet high. What is the height of the attic?
The height of the attic in Mr. Lee's house is 6 1/3
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators like addition, subtraction, multiplication and division. Equations can be linear, quadratic.
Mr. Lee's house has two stories and an attic. Therefore:
Height of entire house = height of first floor + height of second floor + height of attic
The first floor is 9½ feet high, the second floor is 8 3/4 feet high, and the entire house is 24 7/12 feet high
9 1/2 = 19/2; 8 3/4 = 27/4; 24 7/12 = 295/12. Let x represent the height of the attic. Substituting:
x + 19/2 + 35/4 = 295/12
x = 19/3
x = 6 1/3
The height of the attic is 6 1/3
Find out more on equation at: https://brainly.com/question/2972832
#SPJ1
The lengths of two sides of a triangle are 4 inches and 7 inches.
What is a possible length, in inches, of the third side?
A possible length, in inches, of the third side is equal to 4 inches.
What is the triangle inequality theorem?In Euclidean geometry, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than or equal (≥) to the third side of the triangle.
Mathematically, the Triangle Inequality Theorem is represented by this mathematical expression:
b - c < n < b + c
Where:
n, b, and c represent the side lengths of this triangle.
By applying the Triangle Inequality Theorem, the possible side lengths of the triangle's third side is given by:
b - c < n < b + c
7 - 4 < n < 7 + 4
3 < n < 11.
Therefore, n could be 4, 5, 6, 7, 8, 9, and 10 inches.
Read more on Triangle Inequality Theorem here: brainly.com/question/26037134
#SPJ1
What is the answers to 3/4 x 3/4
Answer: 9/16
Step-by-step explanation: Also known as (3/4)^2
1. 3x3/4x4
2. 9/4x4
3. 9/16
You can't simplify anymore because it is in the simplest form already.
This circle iS centered at the origin, and the length of its radius iS 5. What is
the equation of the circle?
Answer:
25
Step-by-step explanation:
The equation of a circle centered at the origin with radius r is given by:
(x^2) + (y^2) = r^2
In this case, the circle is centered at the origin and has a radius of 5, so the equation is:
x^2 + y^2 = 25
Therefore, the equation of the circle is: x^2 + y^2 = 25
Find the 11th term of the geometric sequence 7, 35, 175
11th term of the given geometric sequence is [tex]a_{n} = 7(5^{n-1})[/tex]
What is Geometric sequence?In a Geometric Sequence each term is found by multiplying the previous term by a constant.
Given geometric sequence = 7,35,175
Nth term of the geometric sequence [tex]a_{n}=a_{1} .r^{n-1}[/tex]
[tex]a_{1}=7[/tex]
[tex]r= \frac{35}{7} =5[/tex]
Nth term of the geometric sequence
[tex]a_{n} = 7(5^{n-1})[/tex]
Hence, the Nth term of the given geometric sequence is [tex]a_{n} = 7(5^{n-1})[/tex]
Find the value of each variable provide proofs
The value of x is 18.
The value of y is 15.6.
What is the Pythagorean theorem?Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
We have,
From the figure,
Sin 30 = 9/x
1/2 = 9/x
x = 9 x 2
x = 18
Now,
Using the Pythagorean theorem.
x² = 9² + y²
18² = 81 + y²
y² = 325 - 81
y = √244
y = 15.6
Thus,
x = 18
y = 15.6
Learn more about the Pythagorean theorem here:
https://brainly.com/question/14930619
#SPJ1
What is 4.25 in expanded form?
Answer:
4 + 0.2 + 0.05
Step-by-step explanation:
The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?
360 guests were at the gala
How to determine how many guests were at the gala?Ratio is used to compare two or more quantities. It is used to indicate how big or small a quantity is when compared to another.
Since the ratio of staff to guests at the gala was 3 to 5.
Let S and G represent the number of staff and guests respectively. From the given information, we can write:
S/G = 3/5
S + G = 576
Thus, S = 576 - G
Put S = 576 - G in S/G = 3/5 and solve for G. That is:
(576 - G)/G = 3/5
3G = 5(576 - G)
3G = 2880 - 5G
3G + 5G = 2880
8G = 2880
G = 2880/8
G = 360
Learn more about ratio on:
brainly.com/question/2328454
#SPJ1
(3b + 3)(-b² - 2b + 2)
help fast need done pls.
my brain is playing not smart on me right now
Can someone please help me with this?
Show work please
The closet square inch that is needed to cover the cuboid is 208 inches ( option A)
What is surface area of a cuboid?A cuboid is defined as a three-dimensional shape, that has six rectangular faces, eight vertices and twelve edges.
The surface area of cuboid is given as;
SA = 2(lb+lh+bh)
lb = 33/4 × 4 = 33 in
lh = 23/4 × 4 = 23 in
bh = 23/4 × 33/4 = 759/16
SA = 2( 23+33+759/16)
SA = 46+66+47.44
SA = 207 in
therefore the closet square inches is 208 in²
learn more about surface area of cuboid from
https://brainly.com/question/26403859
#SPJ1
how to calculating rf values?
Answer:
Step-by-step explanation:
Rf values (retention factor values) are used in thin layer chromatography (TLC) to measure the separation of components in a mixture. They are calculated as follows:
Rf = (distance traveled by the solute) / (distance traveled by the solvent)
The distance traveled by the solute is measured from the origin (the spot where the sample was applied) to the final position of the solute on the TLC plate. The distance traveled by the solvent is measured from the origin to the solvent front (the point where the solvent stops moving up the plate).
It is important to note that Rf values are unique to each solute and are dependent on the solvent used, the type of TLC plate, and the conditions of the experiment (such as the temperature and humidity). To ensure accurate comparison of Rf values, it is necessary to standardize the conditions of the experiment.
Calculating Rf values is a crucial step in the interpretation of TLC data, as it allows for the identification of the components of a mixture based on the unique Rf values of each solute.