Function Question - Walking and Rowing

[MacLaddy]

Homework Statement

This is a word problem I have come across in the first chapter of my Calculus class. It is an Algebra review chapter. Please see the attached photo for a visual representation.

"Walking and Rowing: Kelly has finished a picnic on an island that is 200m off shore, as shown in the figure. She wants to return to a beach house that is 600m from the point P on the shore closest to the island. She plans to row a boat to a point on shore x meters from P and then jog along the (straight) shore to the house."

a: Let d(x) be the total length of her trip as a function of x. Graph this function.
b: Suppose that Kelly can row at 2 m/s and jog at 4 m/s. Let T(x) be the total time for her trip as a function of x. Graph y = T(x)
c: Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?

Homework Equations

$$a^2 + b^2 = c^2$$

The Attempt at a Solution

a: I believe I have found the solution to a. Just using the Pythagorean Theorem and some logic I can see that $d(x)=\sqrt{x^2 + 200^2} + (600-x)$ However, I can't see if I can simplify that any further. It doesn't appear that I can, but I am RUSTY, so I wouldn't be surprised.

b: I am having a difficult time wrapping my head around this question. Is it asking me to graph the function of time with x being the same x as in question A? Or am I overlaying this additional function on top of "a's" graph, and then showing where they intersect? Even if that is the case I am having difficulties seeing how to turn this into a function. T(x)=2x+4x, but it would need some other variable to indicate the distance. What am I missing here?

c: Can't get to that without b.

Any information will be greatly appreciated. I've been scratching my head over this one and it is definitely eluding me.

 

[Ray Vickson]

Homework Statement

This is a word problem I have come across in the first chapter of my Calculus class. It is an Algebra review chapter. Please see the attached photo for a visual representation.

"Walking and Rowing: Kelly has finished a picnic on an island that is 200m off shore, as shown in the figure. She wants to return to a beach house that is 600m from the point P on the shore closest to the island. She plans to row a boat to a point on shore x meters from P and then jog along the (straight) shore to the house."

a: Let d(x) be the total length of her trip as a function of x. Graph this function.
b: Suppose that Kelly can row at 2 m/s and jog at 4 m/s. Let T(x) be the total time for her trip as a function of x. Graph y = T(x)
c: Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?

Homework Equations

$$a^2 + b^2 = c^2$$

The Attempt at a Solution

a: I believe I have found the solution to a. Just using the Pythagorean Theorem and some logic I can see that $d(x)=\sqrt{x^2 + 200^2} + (600-x)$ However, I can't see if I can simplify that any further. It doesn't appear that I can, but I am RUSTY, so I wouldn't be surprised.

b: I am having a difficult time wrapping my head around this question. Is it asking me to graph the function of time with x being the same x as in question A? Or am I overlaying this additional function on top of "a's" graph, and then showing where they intersect? Even if that is the case I am having difficulties seeing how to turn this into a function. T(x)=2x+4x, but it would need some other variable to indicate the distance. What am I missing here?

c: Can't get to that without b.

Any information will be greatly appreciated. I've been scratching my head over this one and it is definitely eluding me.

Your d(x) = total distance travelled, rowing + jogging. That is not what the question asks about; it asks about total _time_ and wants you to minimize that. Hints: what is the rowing time in terms of x? What is the jogging time in terms of x? What is the total time?

RGV

 

[SammyS]

Homework Statement

This is a word problem I have come across in the first chapter of my Calculus class. It is an Algebra review chapter. Please see the attached photo for a visual representation.

"Walking and Rowing: Kelly has finished a picnic on an island that is 200m off shore, as shown in the figure. She wants to return to a beach house that is 600m from the point P on the shore closest to the island. She plans to row a boat to a point on shore x meters from P and then jog along the (straight) shore to the house."

a: Let d(x) be the total length of her trip as a function of x. Graph this function.
b: Suppose that Kelly can row at 2 m/s and jog at 4 m/s. Let T(x) be the total time for her trip as a function of x. Graph y = T(x)
c: Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?

Homework Equations

$$a^2 + b^2 = c^2$$

The Attempt at a Solution

a: I believe I have found the solution to a. Just using the Pythagorean Theorem and some logic I can see that $d(x)=\sqrt{x^2 + 200^2} + (600-x)$ However, I can't see if I can simplify that any further. It doesn't appear that I can, but I am RUSTY, so I wouldn't be surprised.

b: I am having a difficult time wrapping my head around this question. Is it asking me to graph the function of time with x being the same x as in question A? Or am I overlaying this additional function on top of "a's" graph, and then showing where they intersect? Even if that is the case I am having difficulties seeing how to turn this into a function. T(x)=2x+4x, but it would need some other variable to indicate the distance. What am I missing here?

c: Can't get to that without b.

Any information will be greatly appreciated. I've been scratching my head over this one and it is definitely eluding me.

Using your d(x),

how much time does it take to travel a distance of $\sqrt{x^2 + 200^2}$ meters at 2 m/s ?

How much time does it take to travel a distance of $600-x$ meters at 4 m/s ?​

Sum those times.

 

[MacLaddy]

Thank you, Ray

I was right in the middle of another question when SammyS posted, and that may have cleared some things up, but thank you for your reply.

SammyS,

Thank you, I was completely failing to make the connection between the land and water functions wrt time. I still am not sure if this is correct, but how does $T(x)= 2\sqrt{x^2+200^2}+4(600-x)$ look?

Something seems wrong. It doesn't graph quite the way I expected it to.

 

[Ray Vickson]

Thank you, Ray

I was right in the middle of another question when SammyS posted, and that may have cleared some things up, but thank you for your reply.

SammyS,

Thank you, I was completely failing to make the connection between the land and water functions wrt time. I still am not sure if this is correct, but how does $T(x)= 2\sqrt{x^2+200^2}+4(600-x)$ look?

Something seems wrong. It doesn't graph quite the way I expected it to.

If I row 100 m at 10 m per minute, how many minutes do I row? is it 100*10 or 100/10?

RGV

 

[Joffan]

$$\text{speed} = \frac{\text{distance}}{\text{time}}$$
Rearrange to give time on the LHS.

 

[MacLaddy]

Ah, thanks everyone for the assistance. The comments above and some sleep helped me to see that solution.
$$T(x)=(\frac{\sqrt{x^2+200^2}}{2})+(\frac{(600-x)}{4})$$