Two ants are at a common point at time t=0, the first ant starts crawling along a straight line at the rate of 4 ft/min. Two minutes later, the second ant starts crawling in a direction perpendicular to that of the first, at a rate of 5 ft/min. How fast is the distance between them changing when the first insect has traveled 12 feet? Thank you!
Answers
Answer 1
Answer:
I looked at this just before going to bed, and it ruined my night, worrying
about it. Then, this morning, when I actually set pencil to paper, it turned
out not to be such a vicious limbic twister.
You DO need to know how to take a chain derivative, though. I must say,
I'm surprised to see this much Calculus at the middle school level. Maybe
it's MY problem, but I just can't see a way to do it without this derivative,
so I'll just show you how I did it:
The ants begin together, and crawl out along the legs of a right triangle.
-- The first one moves at 4 ft/min, so after 't' minutes, he's 4t-ft from
the vertex. The question concerns what happens when this one has
covered 12-ft. That will happen in (12/4) = 3 minutes after he starts out.
-- The second one starts out 2-min later, and moves at 5-ft/min. So,
after 't' minutes, he's 5(t-2) ft from the vertex.
-- The distance between them is the hypotenuse of the right triangle.
The square of the distance is the sum of the squares of the legs:
D² = (4t)² + [ 5(t-2) ]²
We're actually going to need to know how far apart they are after 3 minutes,
so let's do that now.
D² = (4 x 3)² + (5 x 1)² = (12)² + (5)² = 144 + 25 = 169
D = √169 = 13-ft apart, after the first guy has covered 12 feet.
We will need this shortly.
OK. Now, let's simplify the equation for the distance between them,
(as we get ready to differentiate it).
D² = (4t)² + [ 5(t-2) ]²
D² = 16t² + (25) (t² - 4t + 4)
D² = 16t² + 25t² - 100t + 100
D² = 41t² - 100t + 100
The question asks "How fast is the distance between them changing . . . ? ",
and we know that this is the derivative of 'D' with respect to 't' . It's easy to
take the derivative of the right side of the equation, but what about the derivative
of ' D² ' ? That's the part that I'm not so sure about in middle school, so I'll just
give it to you:
(Derivative of D²) is [ (2D) times (derivative of D) ] .
Now we can take the derivative of each side of the equation:
(2D) x (derivative of D) = 82 t - 100 .
derivative of D = (82 t - 100) / (2D)
That's how fast the distance between them is changing at any time 't' .
After 3 minutes, we calculated 'D' earlier. It's 13-ft
derivative of D = [ (82 x 3) - 100 ] / (2 x 13)
= 146 / 26
= 5.615 feet per second.
Are you confident ? Do you trust me?
I'm not confident at all, and I don't trust myself.
The best I can suggest is for you to go through this and look for my mistakes,
or better yet, sit down with your teacher and go through it for mistakes.
That'll help you think about it and understand it, even if my work is wrong.
I looked at this just before going to bed, and it ruined my night, worrying
about it. Then, this morning, when I actually set pencil to paper, it turned
out not to be such a vicious limbic twister.
You DO need to know how to take a chain derivative, though. I must say,
I'm surprised to see this much Calculus at the middle school level. Maybe
it's MY problem, but I just can't see a way to do it without this derivative,
so I'll just show you how I did it:
The ants begin together, and crawl out along the legs of a right triangle.
-- The first one moves at 4 ft/min, so after 't' minutes, he's 4t-ft from
the vertex. The question concerns what happens when this one has
covered 12-ft. That will happen in (12/4) = 3 minutes after he starts out.
-- The second one starts out 2-min later, and moves at 5-ft/min. So,
after 't' minutes, he's 5(t-2) ft from the vertex.
-- The distance between them is the hypotenuse of the right triangle.
The square of the distance is the sum of the squares of the legs:
D² = (4t)² + [ 5(t-2) ]²
We're actually going to need to know how far apart they are after 3 minutes,
so let's do that now.
D² = (4 x 3)² + (5 x 1)² = (12)² + (5)² = 144 + 25 = 169
D = √169 = 13-ft apart, after the first guy has covered 12 feet.
We will need this shortly.
OK. Now, let's simplify the equation for the distance between them,
(as we get ready to differentiate it).
D² = (4t)² + [ 5(t-2) ]²
D² = 16t² + (25) (t² - 4t + 4)
D² = 16t² + 25t² - 100t + 100
D² = 41t² - 100t + 100
The question asks "How fast is the distance between them changing . . . ? ",
and we know that this is the derivative of 'D' with respect to 't' . It's easy to
take the derivative of the right side of the equation, but what about the derivative
of ' D² ' ? That's the part that I'm not so sure about in middle school, so I'll just
give it to you:
(Derivative of D²) is [ (2D) times (derivative of D) ] .
Now we can take the derivative of each side of the equation:
(2D) x (derivative of D) = 82 t - 100 .
derivative of D = (82 t - 100) / (2D)
That's how fast the distance between them is changing at any time 't' .
After 3 minutes, we calculated 'D' earlier. It's 13-ft
derivative of D = [ (82 x 3) - 100 ] / (2 x 13)
= 146 / 26
= 5.615 feet per second.
Are you confident ? Do you trust me?
I'm not confident at all, and I don't trust myself.
The best I can suggest is for you to go through this and look for my mistakes,
or better yet, sit down with your teacher and go through it for mistakes.
That'll help you think about it and understand it, even if my work is wrong.
Answer 2
Answer:
Ok, so I'll be quantizing time here to get you the most accurate result I can give you.
What you have to know is that:
Also:
t=time snapshot (in minutes)
x=distance first ant has travelled
y=distance second ant has travelled (perpendicular to the first ant)
D=distance between both ants
--------------------------------------
Now, when:
t=0, x=0, y=0, D=0
t=1, x=4, y=0, D=4
t=2, x=8, y=0, D=8
t=3, x=12, y=5, D=13
Now we quantize time and find the tangent line (slope) between two points on the distance vs time graph at t=3 and t=3.000001, knowing that distance is represented by feet and time is represented in minutes.
When t=3.000001, x=12.000004, y=5.000005 D=13.00000562...
Now we find the slope on the distance vs time graph between the points (3,13) and (3.000001, 13.00000562).
So, at the point (3,13) on the distance vs time graph, the two ants are moving away from each other at approximately 5.62 feet per minute.
Notations you may require to understand problem in the attachment below.
What you have to know is that:
Also:
t=time snapshot (in minutes)
x=distance first ant has travelled
y=distance second ant has travelled (perpendicular to the first ant)
D=distance between both ants
--------------------------------------
Now, when:
t=0, x=0, y=0, D=0
t=1, x=4, y=0, D=4
t=2, x=8, y=0, D=8
t=3, x=12, y=5, D=13
Now we quantize time and find the tangent line (slope) between two points on the distance vs time graph at t=3 and t=3.000001, knowing that distance is represented by feet and time is represented in minutes.
When t=3.000001, x=12.000004, y=5.000005 D=13.00000562...
Now we find the slope on the distance vs time graph between the points (3,13) and (3.000001, 13.00000562).
So, at the point (3,13) on the distance vs time graph, the two ants are moving away from each other at approximately 5.62 feet per minute.
Notations you may require to understand problem in the attachment below.
Related Questions
you spend $50 on a meal for you and your friends. graph the equation 1.5y+4x=50. x=number of sandwiches bought and x=number of drinks bought.
On her first quiz in social studies, Meg answered 92% of the questions correctly. On her second quiz, she answered 27 out of 30 questions correctly. On which quiz did Meg have the better score?
Which mixed number has the same value as 3.75?A 3 & 1/2B 3 & 5/8C 3 & 3/4D 3 & 7/10
Time(weeks) Lake Depth(feet)0 346.02 344.85 343.07 341.810 340.012 338.8An environmental agency studying the effects of drought on a lake found that the water level decreased at a constant rate during the first few months of the drought. When the agency began its research, the depth of the lake was 346 feet. The table represents the agency’s susequent observations about the lake's depth.Which equation represents the depth of the lake, y, based on the number of weeks passed, x?y = -0.6x + 346y = -1.2x + 346y = -0.6xy = -1.2x
Solve a real world problem that can be represented by the expression (-3)(5)+10
On her first quiz in social studies, Meg answered 92% of the questions correctly. On her second quiz, she answered 27 out of 30 questions correctly. On which quiz did Meg have the better score?
Which mixed number has the same value as 3.75?A 3 & 1/2B 3 & 5/8C 3 & 3/4D 3 & 7/10
Time(weeks) Lake Depth(feet)0 346.02 344.85 343.07 341.810 340.012 338.8An environmental agency studying the effects of drought on a lake found that the water level decreased at a constant rate during the first few months of the drought. When the agency began its research, the depth of the lake was 346 feet. The table represents the agency’s susequent observations about the lake's depth.Which equation represents the depth of the lake, y, based on the number of weeks passed, x?y = -0.6x + 346y = -1.2x + 346y = -0.6xy = -1.2x
Solve a real world problem that can be represented by the expression (-3)(5)+10
Answers
Use the pythagorean theorem.
The foot of the ladder is 4 feet away from the building (leg 1)
The height of the building is leg 2
The ladder, which would be at an angle, is 12 feet. So our hypotenuse would be 12. The equation for this would be a²+4²=12²
We can then begin to solve
a² + 16 = 144
a² = 128
a=11.31
So, our answer is: The top of the ladder reaches the building at 11.31 feet up
Answers
My first reaction is to say: "In that case, you ought to get busy."
(1,000,000 hours) x (1 day / 24 hours) x (1 year / 365.25 days) =
(1,000,000 x 1 x 1) / (24 x 365.25) years =
114.08 years .
A few human beings live that long.
Most don't.
Answers
a) To find amount collected in each category - you must understand that the numbers inside the circle represent an angle (in units of degrees). Since we know a circle is composed of 360 degrees, we know each other portions as a ratio of that full circle
for simplicity sake, i will use "d" to represent "degrees"
Shirts = (140d)/(360d)*($7200) = $2800
Pants = (80d)/(360d)*($7200) = $1600
Other = (75d)/(360d)*($7200) = $1500
Shoes = (360d - 140d - 80d - 75d)/(360d)*($7200) = $1300
Total = $7200 [OK]
b) Knowing long sleeve shirts are to be 1/4 of the shirts sales, find the angle which represents in on the pie graph:
angle (long sleeve shirts) = 1/4*(140d) = 35d
c) Assuming ratios to maintain over the next $900 of sales
Pant sales (in the next $900 of total sales) = ($900)*(80d/360) = $200
hope that helps
for simplicity sake, i will use "d" to represent "degrees"
Shirts = (140d)/(360d)*($7200) = $2800
Pants = (80d)/(360d)*($7200) = $1600
Other = (75d)/(360d)*($7200) = $1500
Shoes = (360d - 140d - 80d - 75d)/(360d)*($7200) = $1300
Total = $7200 [OK]
b) Knowing long sleeve shirts are to be 1/4 of the shirts sales, find the angle which represents in on the pie graph:
angle (long sleeve shirts) = 1/4*(140d) = 35d
c) Assuming ratios to maintain over the next $900 of sales
Pant sales (in the next $900 of total sales) = ($900)*(80d/360) = $200
hope that helps
If It is in fact (b), then it's 90 degrees
Answers
Mary will have 5 pets if she buys a bird
Answers
All you need to do is multiply 9 by 96. Your answer is 864
Answers
i think the maximum length of the figure would be : 6The total blocks that exist to form should be 8we know that 3 is used for the height , so the remaining will be 5.but we can use the first blocks as the foundation for the additional length. The figure would look like this :OOOOOOOO ---------> total of 8 blocks
Other Questions