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Problem 57 Hard Difficulty

A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon.

(a) Express the distance $ s $ between the lighthouse and the ship as a function of $ d $, the distance the ship has traveled since noon; that is, find $ f $ so that $ s = f(d) $.
(b) Express $ d $ as a function of $ t $, the time elapsed since noon; that is, find $ t $ so that $ d = g(t) $.
(c) Find $ f \circ g $. What does this function represent?

Answer

(a) From the figure, we have a right triangle with legs 6 and $d$, and hypotenuse $s$.
By the Pythagorean Theorem, $d^{2}+6^{2}=s^{2} \Rightarrow s=f(d)=\sqrt{d^{2}+36}$.
(b) Using $d=r t,$ we get $d=(30 \mathrm{km} / \mathrm{h})(t \text { hours })=30 t$ (in $\mathrm{km}$ ). Thus,
$d=g(t)=30 t$.
(c) $(f \circ g)(t)=f(g(t))=f(30 t)=\sqrt{(30 t)^{2}+36}=\sqrt{900 t^{2}+36} .$ This function represents the distance between the lighthouse and the ship as a function of the time elapsed since noon.

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Discussion

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CR

Cam R.

September 21, 2020

Is it standard to use kilometers to solve this?

LP

Lindsey P.

September 21, 2020

Hey Cam The metre is the Standard International (SI) unit of length. ... are particularly good to use because you can measure its dimensions and then calculate the number

ST

Samantha T.

September 21, 2020

What is a Pythagorean theorem?

HC

Howie C.

September 21, 2020

Well in mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the

AG

Alex G.

September 21, 2020

Can someone explain what the square root is?

NH

Nadia H.

September 21, 2020

I know this one! A number which produces a specified quantity when multiplied by itself.

Video Transcript

Okay, let's draw a picture of the situation. So we have the shoreline and we have the lighthouse, and we know that the ship is six kilometers out from the shoreline and it's traveled a distance of D since noon. So here's the ship and we want to find the distance from the ship to the lighthouse. More calling. That s so we have a right triangle. And so if we want to find s as a function of D, we can use the Pythagorean theorem to relate s and D So we have s squared equals six squared plus b squared and that is s squared equals 36 plus D squared. We can square root both sides and we get s equals the square root of 36 plus d squared. We don't need the plus or minus because we know it's just a positive quantity. Since we're talking about a length, Okay, now it's do part two, and in this part, we want to find distance as a function of time. We know the relationship, distance equals rate, times, time, and we were told in the problem that the rate of the ship is 30 kilometers per hour, so the distance will be 30 times teeth. Now, for part C, we're told to let our distance from the ship to the lighthouse function that we called s earlier recall Word told to let that be f and then the distance as a function of time that we just found. That was our d were asked to call that G and we want to find f of G. So G is going inside f and we get the square root of 36 plus 30 t squared. And if we want to simplify that, that would be the square root of 36 plus 900 t squared. And what that represents is the distance from the lighthouse to the ship as a function of time.