1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ymorist [56]
3 years ago
12

The profile of the cables on a suspension bridge may be modeled by a parabola. The central span of the bridge is 1210 m long and

128 m high. The parabola y equals 0.00035 x squared gives a good fit to the shape of the​ cables, where StartAbsoluteValue x EndAbsoluteValue less than or equals 605​, and x and y are measured in meters. Approximate the length of the cables that stretch between the tops of the two towers.
Mathematics
1 answer:
brilliants [131]3 years ago
7 0

Answer:

The approximated length of the cables that stretch between the tops of the two towers is 1245.25 meters.

Step-by-step explanation:

The equation of the parabola is:

y=0.00035x^{2}

Compute the first order derivative of <em>y</em> as follows:

 y=0.00035x^{2}

\frac{\text{d}y}{\text{dx}}=\frac{\text{d}}{\text{dx}}[0.00035x^{2}]

    =2\cdot 0.00035x\\\\=0.0007x

Now, it is provided that |<em>x </em>| ≤ 605.

⇒ -605 ≤ <em>x</em> ≤ 605

Compute the arc length as follows:

\text{Arc Length}=\int\limits^{x}_{-x} {1+(\frac{\text{dy}}{\text{dx}})^{2}} \, dx

                  =\int\limits^{605}_{-605} {\sqrt{1+(0.0007x)^{2}}} \, dx \\\\={\displaystyle\int\limits^{605}_{-605}}\sqrt{\dfrac{49x^2}{100000000}+1}\,\mathrm{d}x\\\\={\dfrac{1}{10000}}}{\displaystyle\int\limits^{605}_{-605}}\sqrt{49x^2+100000000}\,\mathrm{d}x\\\\

Now, let

x=\dfrac{10000\tan\left(u\right)}{7}\\\\\Rightarrow u=\arctan\left(\dfrac{7x}{10000}\right)\\\\\Rightarrow \mathrm{d}x=\dfrac{10000\sec^2\left(u\right)}{7}\,\mathrm{d}u

\int dx={\displaystyle\int\limits}\dfrac{10000\sec^2\left(u\right)\sqrt{100000000\tan^2\left(u\right)+100000000}}{7}\,\mathrm{d}u

                  ={\dfrac{100000000}{7}}}{\displaystyle\int}\sec^3\left(u\right)\,\mathrm{d}u\\\\=\dfrac{50000000\ln\left(\tan\left(u\right)+\sec\left(u\right)\right)}{7}+\dfrac{50000000\sec\left(u\right)\tan\left(u\right)}{7}\\\\=\dfrac{50000000\ln\left(\sqrt{\frac{49x^2}{100000000}+1}+\frac{7x}{10000}\right)}{7}+5000x\sqrt{\dfrac{49x^2}{100000000}+1}

Plug in the solved integrals in Arc Length and solve as follows:

\text{Arc Length}=\dfrac{5000\ln\left(\sqrt{\frac{49x^2}{100000000}+1}+\frac{7x}{10000}\right)}{7}+\dfrac{x\sqrt{\frac{49x^2}{100000000}+1}}{2}|_{limits^{605}_{-605}}\\\\

                  =1245.253707795227\\\\\approx 1245.25

Thus, the approximated length of the cables that stretch between the tops of the two towers is 1245.25 meters.

You might be interested in
etermine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. 3 2 x
nata0808 [166]

Answer:

(32/15, -2/5)

Step-by-step explanation:

(1) ³/₂x   -    2y =  4

(2)   x  +    ⅓y =  2

(3)  3x   -     4y =  8              Multiplied (1) by 2

(4)  3x  +       y =  6              Multiplied (2) by 3

                  5y = -2              Subtracted (3) from (4)

(5)                 y = -⅖             Divided each side by 5

       x + ⅓(-⅖) =  2              Substituted (5) into (2)

       x  -  ²/₁₅    =  2

       x              =  2 + ²/₁₅     Added ²/₁₅ to each side

      x              = 32/15

The system of equations has only one solution: (32/15, -2/5).

3 0
3 years ago
The base sales tax rate in Florida is 6%,
Slav-nsk [51]

Answer:

$18.43

Step-by-step explanation:

19.61 decrease 6% =

19.61 × (1 - 6%) = 19.61 × (1 - 0.06) = 18.4334

5 0
2 years ago
if p(x) = x2 – 1 and q(x)=5(x-1), which expression is equivalent to (p – q)(x)? a: 5(x – 1) – x2 – 1 b: (5x – 1) – (x2 – 1) c: (
zavuch27 [327]
P(x) = x^2 - 1
q(x) = 5(x - 1)
(p - q)(x) = (x^2 - 1) - 5(x - 1)
7 0
3 years ago
Read 2 more answers
how fast would 1.5 times the faster rowing speed be? before you solve tell the number of decimal places on your answer PLZ HELP
AleksAgata [21]

How far are you rowing?

If your rowing speed is 1.5 times faster your going about 16 miles up to 2 hours in the river

Sorry if this is wrong-

~zaro~

5 0
3 years ago
What is the area , in square meters of the shaded part of the rectangle below
Licemer1 [7]

Answer:

123.5 m^2

Step-by-step explanation:

The area of the whole rectangle is 9 x 19 = 171 m^2

The area of the region that isn't shaded is (5 x 19)/2 = 47.5 m^2 because the height of the triangle is 5 (9 - 4) and the area of a triangle is

(height x length)/2

171 - 47.5 = 123.5 m^2

4 0
3 years ago
Other questions:
  • Elevator 1 moved up 10 feet from the ground level. Its position is labeled as +10. Elevator 2 moved down 5 feet from the ground
    9·1 answer
  • Pairs of angles nonadjacent formed when a transversal intersects two lines so that they lie on the same side of the transversal
    13·1 answer
  • A prism is completely filled with 96 cubes that have edge length of 12 cm. What is the volume of the prism?
    5·2 answers
  • 4-30. Micron Technology has sales offices located in four
    8·1 answer
  • PLSSSSSSS HELPPPPPPP I WILL GIVE BRAINLIESTTTTTTTTTT!!!!!!!!!!!!!!!!!!!!!PLSSSSSSS HELPPPPPPP I WILL GIVE BRAINLIESTTTTTTTTTT!!!
    6·2 answers
  • Evaluate 8^5/3<br> Helplpppp
    10·2 answers
  • katy desea alquilar un barca y gastar por menos $54. El barco cuesta $7 por hora y katy tiene un cupom de descuento de $9 dolare
    5·1 answer
  • In the parallelogram shown, find x in terms of u and v. ​
    6·1 answer
  • Which of the following sets of numbers could represent the three sides of a triangle?
    10·1 answer
  • Figure 22: Find the total yellow area.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!