MHB Beverly's question at Yahoo Answers regarding related rates

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    Related rates
AI Thread Summary
The discussion revolves around a calculus problem involving related rates, where a trooper measures the distance to a car moving south. Using the Pythagorean theorem, the relationship between the distances is established, leading to the differentiation of the equation with respect to time. The solution reveals that the speed of Amanda's car is 90 feet per second, calculated by substituting known values into the derived formula. The response encourages further engagement by inviting others to post similar problems for discussion. The thread effectively addresses the calculus question while fostering a collaborative learning environment.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Calculus Homework Question - related rates help?

Amanda is driving her car south on Interstate 95. A Virginia State Trooper is parked 90 feet west of the interstate, and aims his radar at the car after it passes him. He finds the distance to Amanda's car from his position is 150 feet and the distance separating them is increasing at the rate of 72 feet per second. Find the speed of the car in feet per second.

Here is a link to the question:

Calculus Homework Question - related rates help? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello Beverly,

The first thing I would do is draw a diagram representing the scenario:

34sfitx.jpg


$T$ represents the trooper's position, $C$ represents the position of Amanda's car on I-95, $x$ represents the distance Amanda has traveled since passing the trooper, and $h$ represents the distance between Amanda and the trooper. All distances are in feet.

Now, we are ultimately being asked to find $$\frac{dx}{dt}$$, and we are given $$\left.\frac{dh}{dt}\right|_{h=150}=72\,\frac{\text{ft}}{\text{s}}$$.

What we need then, is a relationship between $x$ and $h$, and fortunately, we have what we need via Pythagoras:

$$x^2+90^2=h^2$$

Now, implicitly differentiating with respect to time $t$, we find:

$$2x\frac{dx}{dt}=2h\frac{dh}{dt}$$

and solving for $$\frac{dx}{dt}$$, we find:

$$\frac{dx}{dt}=\frac{h}{x}\frac{dh}{dt}$$

Using the Pythagorean relation, we find:

$$x(h)=\sqrt{h^2-90^2}$$

and so we have:

$$\frac{dx}{dt}=\frac{h}{\sqrt{h^2-90^2}}\frac{dh}{dt}$$

and finally, we may compute:

$$\left.\frac{dx}{dt} \right|_{h=150}=\frac{150}{\sqrt{150^2-90^2}}\left.\frac{dh}{dt} \right|_{h=150}=\frac{5}{4}\cdot72\, \frac{\text{ft}}{\text{s}}=90\, \frac{\text{ft}}{\text{s}}$$

To Beverly and any other guests viewing this topic, I invite and encourage you to post other related rates problems in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Back
Top