The main cables of a suspension bridge are ideally parabolic. the cables over a bridge that is 400 feet logn are attached to towers that 100 feet tall. the lower point of the cable is 40 feet abouve the bdrige. find the equation that can be used to model the cables.
Answers
Let the vertex be on the y axis, exactly, at the point (0, 40)
Another point of this parabola is (200, 100), as can be checked from the figure.
The vertex form of the equation of a parabola is :
[tex]y=a(x-h)^{2}+k [/tex], where (h, k) is the vertex of the parabola,
replacing (h, k) with (0, 40), we have:
[tex]y=ax^{2}+40[/tex]
to find a, we substitute (x, y) with (200, 100):
[tex]100=a200^{2}+40[/tex]
40,000a=60
a=60/40,000=3/(2,000)
So, the equation of the parabola is [tex]y= \frac{3}{2000} x^{2}+40[/tex]
The equation to model the suspension cables of a bridge, where the lowest point of the cable is 40 feet above the bridge, and the towers are 100 feet tall and 200 feet from the center, is y = -0.0015x^2 + 60.
Explanation:To find the equation that models the suspension cable of a bridge, we use the properties of a parabolic shape. Since the cable is attached to towers that are 100 feet tall and the lowest point of the cable is 40 feet above the bridge, we know the vertex of the parabola is 60 feet (100 - 40) below the top of the towers.
Let's define the coordinate system with the origin at the lowest point of the cable. Then the towers are at (-200, 60) and (200, 60) because the bridge is 400 feet long, so each tower is 200 feet horizontally from the center. The parabolic equation takes the general form y = ax^2 + bx + c. Because the vertex is at (0, 60), c = 60.
Using the points (-200, 60), we can substitute into the parabolic equation and write a system to solve for a and b. Since the parabola is symmetric, b = 0. The system becomes 60 = a(-200)^2 + 60, which simplifies to a = -60/40000.
Thus, the equation to model the cables is y = -60/40000x^2 + 60, or simplified, y = -0.0015x^2 + 60.
Related Questions
A junior basketball has a diameter of approximately 7 in., and a regulation basketball has a diameter of approximately 9.5 in. about how many times as great is the volume of the regulation basketball as the volume of the junior basketball?
Answers
the ratio of the volumes will be 9.5^3 / 7^3
= 2.5 to nearest thousandth
So the volume the regulation ball is 2.5 times the volume of the junior one.
1. A ladybug lands on the end of a pinwheel that is spinning at a steady speed. The equation y = –7 sin (0.4πx) + 11 gives the height of the ladybug y, in centimeters, above the ground x seconds after landing. What is the diameter of the pinwheel
Answers
Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth. −6y2 − 9y = −1
Answers
[tex]y= \dfrac{-9+ \sqrt{9^2-4*-6} }{2*6}, \dfrac{-9-\sqrt{9^2-4*-6}}{2*6} [/tex]
[tex]y= \dfrac{-9+ \sqrt{105} }{12}, \dfrac{-9- \sqrt{105} }{12} [/tex]
How can an expression or process be determined for an arithmetic sequence?
Answers
[tex]a_n=a_1+d(n-1)[/tex]
where
an=nth term
a1=first term
d=common differnce, or how much each term increases by
n=which term
example
1,3,5, etc
first term is 1
common difference is 2 because it increases by 2 each time
so the formula would be
[tex]a_n=1+2(n-1)[/tex]
if we had
5,3,1
first term s 5
common difference is -2 since it goes up by -2 each time
so formula is
[tex]a_n=5-2(n-1)[/tex]
hope this helps
Joe the trainer has two solo workout plans that he offers his clients: Plan A and Plan
b. Each client does either one or the other (not both). On Monday there were 2 clients who did Plan A and 3 who did Plan
b. On Tuesday there were 4 clients who did Plan A and 8 who did Plan
b. Joe trained his Monday clients for a total of 7 hours and his Tuesday clients for a total of 17 hours. How long does each of the workout plans last?
Answers
we are assuming personal training for each client.
i)
"On Monday there were 2 clients who did Plan A and 3 who did Plan"
the total time spent is : 2*a + 3*b =2a+3b
ii)
"On Tuesday there were 4 clients who did Plan A and 8 who did Plan B"
the total time spent was 4*a+8*b=4a+8b
iii) "Joe trained his Monday clients for a total of 7 hours"
so 2a+3b = 7
iv)
"Joe trained his Tuesday clients for a total of 17 hours"
so 4a+8b=17
v) thus we have the following system of equations:
2a+3b = 7
4a+8b=17
multiply the first equation by -2, and then add both equations, to eliminate a:
-4a-6b=-14
4a+8b=17
-------------------
2b=3, so b=3/2
2a+3b = 7
2a+3(3/2)=7
2a+9/2=7
multiply by 2:
4a+9=14
4a=5
a=5/4
Answer :
Plan A lasts 5/4=1.25 h
Plan B lasts 3/2=1.5 h
The population of current statistics students has ages with mean muμ and standard deviation sigmaσ. samples of statistics students are randomly selected so that there are exactly 4242 students in each sample. for each sample, the mean age is computed. what does the central limit theorem tell us about the distribution of those mean ages?
Answers
From each sample of 42, we can compute a mean value of [tex]\Bar{x}[/tex].
We do not know the distribution for any sample.
According to the Central Limit Theorem, the distribution of the sample means will follow a normal distribution, regardless of the distribution of the individual samples.
Answer:
The distribution of sample means is normally distributed, and [tex]\Bar{x} \Rightarrow \mu[/tex],
Suppose the vertex of a parabola is in the first quadrant and the parabola opens upwards. What can be determined about the value of a and the discriminant?
Answers
that is the graph of [tex]f(x)=a x^{2} +bx+c[/tex], where a is not 0.
from a, b and c we can derive the following informations about the shape of a parabola:
if a>0, the parabola opens upwards.
if a<0, the parabola opens downwards.
Consider the discriminant [tex]D= b^{2} -4ac[/tex]
If D>0, the parabola intersects the x-axis at 2 points.
If D=0, the parabola intersects the x-axis at 1 point.
If D<0, the parabola does not intersect the x axis.
"the vertex of a parabola is in the first quadrant and the parabola opens upwards."
the vertex is in the first quadrant means that the vertex is above the x-axis, and it opens upwards, so the parabola does not intersect the x-axis.
This means that:
Answer: a>0, the discriminant D<0
Final answer:
A parabola in the first quadrant opening upwards implies a positive 'a' value and a discriminant that, if not negative, yields real roots with positive values.
Explanation:
When a parabola has its vertex in the first quadrant and it opens upwards, we can determine specific values for a and the discriminant. The coefficient 'a' in the quadratic equation ax²+bx+c = 0 must be positive for the parabola to open upwards. Concerning the discriminant (calculated as b²-4ac), if the vertex is in the first quadrant, the parabola either does not intersect the x-axis at all (discriminant < 0), or it intersects the x-axis at one point (discriminant = 0) or two points (discriminant > 0) that both have positive x values.
The discriminant plays a key role in determining the nature of the roots of the quadratic equation. For quadratic equations constructed on physical data, they usually have real roots. Practical applications often deem the positive roots significant.
How do you solve equations for indicated variables?
ax+r=7
Solving for x?
Answers
ax=-r+7
then you would divide both sides by A
so then X = A/(-r+7)
and now you can plug in for A and R to solve
The solution is x = (7 - r) / a.
To solve the equation ax + r = 7 for x, follow these steps:
Subtract r from both sides of the equation:
ax + r - r = 7 - r
This simplifies to:
ax = 7 - r
Divide both sides by a:
x = (7 - r) / a
Substitute the value of x back into the original equation to ensure it is correct.
A carnival game allows a group of players to each draw and keep a marble from a bag. The bag contains 5 gold marbles, 25 silver marbles, and 70 red marbles.
A player wins a large prize for drawing a gold marble and a small prize for drawing a silver marble. There is no prize for drawing a red marble.
At the start of the game, the probability of winning a large prize is 0.05 and the probability of winning a small prize is 0.25.
1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?
2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?
3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?
Answers
25 silver marbles, and
70 red marbles.
------------------------------
100 total marbles
large prize: drawing a gold marble
small prize: drawing a silver marble.
At the start of the game,
probability of winning a large prize = positive outcoumes / total possible outcomes = 5 gold marbles / 100 total marbles = 0.05
probability of winning a small prize = positive outcomes / total possible outcomes = 25 silver marbles / 100 total marbles = 0.25.
1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?
Answer:
numer of silver marbles / number of total marbles = (25 -1 ) / (100 - 1) = 24 / 99 ≈ 0.24
2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?
Answer:
First player: 0.25
Second player: 5 gold marbles / ( 100 - 1) total marbles = 5 /99 ≈ 0.0505
Third player: 5 gold marbles / (99 - 1) total marbles = 5 / 98 ≈ 0.051
Fourth player: 5 gold marbles / ( 98 - 1) total marbles = 5 / 97 ≈ 0.0515
So, the probability of winning a big prize increases as more balls different of gold marbles are extracted from the bag, and so, in this case, the fourth player has a greater chance to win a large prize.
3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?
Answer: All the players would have equal chance of winning a prize if the balls were replaced in the bag after each play.
You invest $500 in an account with an annual interest rate of 1.1%, compounded continuously. How much money is in the account after 15 years? Round your answer to the nearest whole number.
Answers
A=p e^rt
A future value?
P present value 500
E constant
R interest 0.011
T time 15 years
A=500×e^(0.011×15)
A=589.7
Solve the system by the elimination method.
x + y - 6 = 0
x - y - 8 = 0
When you eliminate y , what is the resulting equation?
Answers
2x-14=0
2x=14
x=7, which makes x+y=6 become:
7+y=6
y=-1, so the solution to the system of equations is the point:
(7, -1)
Answer: 2x = 14
Step-by-step explanation:
Solving the equation us in elimination method,
x + y - 6 = 0...1
x - y - 8 = 0...2
From 1,
x+y = 6...3
x-y = 8...4
To eliminate y, we will add equation 3 and 4 since both the signs attached to y are different.
2x=6+8
2x = 14 (This will be the resulting equation)
To get the variables x, we will divide both sides of the resulting equation by 2
x = 14/2
x = 7
Substituting x = 7 into eqn 3
7 + y = 6
y = -1
A spherical scoop of ice cream is placed on top of a hollow ice cream cone. the scoop and cone have the same radius. the ice cream melts completely and it fills the cone to the top. how many times greater is the height of the cone than the radius of the cone?
Answers
The volume of the empty cone is
V₁ = (1/3) π r²h
The volume of the sphere is
V₂ = (4/3) π r³
Because the melted ice cream completely fills the cone, therefore
V₁ = V₂
(1/3) π r² h = (4/3) π r³
Divide each side by (1/3) π r².
h = 4r
Answer:
The height of the cone is 4 times greater than the radius f the cone.
50 POINTS!!!!!!!!!
Find a number that is between 7/11 and 0.75, theres a line on top of 75
77/99
23/33
25/44
76/99
To convert 0.9 to a fraction, Lauren wrote n=0.9 as her first step. For her second step, what should she multiply both sides of that equation by?
10
100
1000
10000
Which number is between 1/7 and 0.2 ?
9/35
6/35
2/9
1/9
Which fraction shows 5.2 as the quotient of two integers?
11/2
21/5
26/5
51/10
Answers
2) 10
3) 6 / 35
4) 26 / 5
Of five letters (a, b, c, d, and e), two letters are to be selected at random. how many possible selections are there
Answers
WHICH ONE IS IT?////
Answers
______________________________________________
A system of linear equations includes the line that is created by the equation y=0.5x-1 and the line through the points (3, 1) and (–5, –7), shown below.
What is the solution to the system of equations?
a. (–6, –4)
b. (0, –1)
c. (0, –2)
d. (2, 0)
Answers
Answer: Solution is,
d. (2, 0)
Step-by-step explanation:
Since, the equation of line that passes through points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is,
[tex](y-y_1)=\frac{x_2-x_1}{y_2-y_1}(y-y_1)[/tex]
Thus, the equation of line through the points (3, 1) and (–5, –7) is,
[tex](y-1)=\frac{-7-1}{-5-3}(x-3)[/tex]
[tex](y-1)=\frac{-8}{-8}(x-3)[/tex]
[tex]y - 1 = x - 3[/tex]
[tex]\implies y = x - 2------(1)[/tex],
Equation of second line is,
[tex]y = 0.5x - 1 -----(2)[/tex],
By equation (1) and (2),
x - 2 = 0.5x - 1 ⇒ 0.5x = 1 ⇒ x = 2,
From equation (1),
We get, y = 0,
Hence, the solution of line (1) and (2) is (2,0).
Kite DCFE is inscribed in circle A shown below:
If the measure of arc DEF is 266°, what is the measure of ∠DEF?
Numerical Answers Expected!
Answers
the arcDEF is the intercepted arc.
Answer:
133 degrees
Step-by-step explanation:
Given is a picture of a kite DCFE inscribed in circle A
The vertices of the kite D, C, E and F lie on the circle.
A is the centre.
Arc DEF = 266 degrees
This means the subtended angle of arc DEF at the centre = Angle DAF = 266
By theorem on circles, we have central angle of any arc is twice the angle subtended by the arc on the remaining part of the circumference
Hence
Angle DEF = 1/2 angle DAF
i.e. DEF =[tex]\frac{266}{2} =133[/tex]
133 degrees
Find the coordinates of point Q that lies along the directed line segment from R(-2, 4) to S(18, -6) and partitions the segment in the ratio of 3:7.
Please help!!
Answers
thus then
[tex]\bf \qquad \textit{internal division of a line segment}\\\\ R(-2,4)\qquad S(18,-6)\qquad ratio1=3\qquad ratio2=7\qquad 3:7\\ \quad \\ \quad \\ \cfrac{RQ}{QS}=\cfrac{ratio1}{ratio2}\implies \cfrac{R}{S}=\cfrac{3}{7} \implies 7R=3S \\\\\\ 7(-2,4)=3(18,-6)[/tex]
[tex]\bf {{ Q=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}}\\ \quad \\ \qquad thus\qquad \\ \quad \\ Q=\left(\cfrac{(7\cdot -2)+(3\cdot 18)}{3+7}\quad ,\quad \cfrac{(7\cdot 4)+(3\cdot -6)}{3+7}\right)[/tex]
Assume the birth of a boy or a girl is equally likely. The probability that a single child is born a girl is 1/2. What is the probability that the next child born to the same familiy will also be a girl?
Answers
probability is 1/4 (b)
Step-by-step explanation:
Write the equation in vertex form
f (x)= x^2-10x+16
Answers
In final terms, it becomes (x-5)^2+16-25=(x-5)^2-9.
What is the answer to this?
Answers
5/8 = x+7/ 2
We need to start by doing cross multiply
(8)(2) = (x+7)*5
16 = 5x + 35
5x = 35 - 16
5x =-19
x = -19/5
The correct answer is -19/5
--------------------------------------------------------------------------
The mistake is easy to see.
Instead of doing 35 - 16 { they did 35+16 which is wrong because you are transferring 16 to the opposite so you must change the positive sign as well.}
I hope that helps, please let me know if you have further questions about my answer.
(15 POINTS) A card is drawn from a deck of 52. What is the probability of drawing either a diamond or a seven?
A) 6/13
B) 17/52
C) 19/52
D) 4/13
Answers
Answer:
The correct answer is 4/13
Step-by-step explanation:
The events "drawing a diamond or a seven" are inclusive events since there is a seven of diamonds. Follow the rule for inclusive events.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Hope this helps! :)
Use the pythagorean theorem to find the distance between x(7,11) and y(-1,5)..
Answers
a²+b²=c²
the distance bewteen the points is the hyptohuse
the legs are the distances between th x and y values
so
(7,11) and (-1,5)
distances between the x values are 1 leg
that is from 7 and -1, a distance of 8 units
distances betweeen the y values are the other leg
that is from 11 to 5
that is 6 units
a²+b²=c²
8²+6²=c²
64+36=c²
100=c²
sqrt both sides
10=c
the distance between them is 10 units
How to factor out the greatest common factor in a polynomial?
Answers
To find something COMMON, we must at least have 2 things!
So , we can find the greatest common factor of 2 polynomials!
Final answer:
To factor out the GCF in a polynomial, identify the highest common factor, write it outside the parentheses, divide each term by the GCF, and write the quotients inside the parentheses.
Explanation:
To factor out the greatest common factor (GCF) in a polynomial, follow these steps:
First, identify the highest common factor that is present in each term of the polynomial.Write down this factor outside of a set of parentheses.Divide each term of the polynomial by the GCF, and place the resulting quotient inside the parentheses. This step can be seen as dividing both sides by the same factor to turn polynomial terms into integers, if that is easier to understand.Check your answer to see if it simplifies further and whether it is reasonable.For example, for the polynomial 6x³ + 9x², the GCF is 3x2. Factoring out the GCF gives us:
3x²(2x + 3)
The products inside the parentheses are the result of dividing the original terms by the GCF. Remember, by finding the GCF, we simplify the algebra and may check the work by expanding the factored form back out to verify it equals the original polynomial.
If a car is $27,000 and loses 15% of its value each year what will be the value in 5 years
Answers
10%- $2,700.
5%- $1,350.
$2,700+ $1,350 = $4,050.
$4,050 × 5 = $20,250.
$27,000 - $20,250 = $6,750.
the value will be $6,750.
Determine the slope and y-intercept of the line.
y = 5x + 4
a.
Slope = 4, y-intercept is (0, 5)
c.
Slope = 5, y-intercept is (0, 4)
b.
Slope = -5, y-intercept is (0, 4)
d.
Slope = 4, y-intercept is (0, -5)
Please select the best answer from the choices provided
A
B
C
D
Answers
Prism M and pyramid N have the same base area and the same height. Cylinder P and prism Q have the same height and the same base perimeter. cone Z has the same base area as cylinder Y, but its height is three times the height of cylinder Y. Which two figures have the same volume?
Choices:
Prism M
Cylinder p
Cone Z
And
Pyramid N
Prism Q
Cylinder Y
Answers
Сone Z and Cylinder Y have the same volume.
K is the midpoint of line segment lm. the coordinates of k are (5, 12) and the coordinates of l are (2, 6), find the coordinates of m.
Answers
L = (2,6)
M = (x2,y2)
midpoint = (x1 + x2)/2 , (y1 + y2) / 2
m = (2 + x2) / 2 , (6 + y2)/2
(2 + x) / 2 = 5
2 + x = 5 * 2
2 + x = 10
x = 10 - 2
x = 8
(6 + y) / 2 = 12
6 + y = 12 * 2
6 + y = 24
y = 24 - 6
y = 18
so M = (8,18) <==
Use the graph below for this question:
graph of parabola going through negative 3, negative 3 and negative 4, negative 1.
What is the average rate of change from x = −3 to x = −4?
3
4
−3
−2
Answers
that is just the slope between (-3,-3) and (-4,-1)
slpe between (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1)
so slope between (-3,-3) and (-4,-1) is (-1-(-3))/(-4-(-3))=(-1+3)/(-4+3)=2/-1=-2
the average rate of change is -2
________ probability is used when we know the number of possible outcomes of the event of interest and the total number of possible outcomes in the sample space.