Hey so I need help on a problem from an assignment that my teacher won't help at all.
-WRITE IN COMPLETE SENTENCES-
"An oil strikes a sand bar that rips a hole in the hull of the ship. Oil begins leaking out of a tanker with spilled oil forming an approximate circle around the tanker."
The radius of the circle is increasing at a rate of 2.2ft.
a) write the area of the circle of oil as a function of radius.
b) Write tge radius of the circle as a function of time.
c)Find the radius of the circle of spilled oil after 2 hrs? what's the radius of the circle after 2.5 hrs?
d) use the result in part(c) to determine the area of the circle after 2hrs and 2.5 hrs.
e) write the function that represents area of the circle of spilled oil as a function of time.
f)Use the result of (e) to determine the area of the result of the circle after 2hrs and 2.5 hrs.
g) calculate the average rate of change of the area of the circle from 2hrs to 2.5 hrs.
h)Calculate the average rate of change of the circle from 5hrs to 7.5 hrs.
i)based on the results obtained in parts (g) and (h),what's happening to the average rate of change of the area of the circle as time passes?
j)If the oil tanker is 150yrds from shore,when will the oil spill first reach the shoreline?(1yd=3ft)
k)How long will it be until 6miles of shore line is contaminated with oil?(1mile=580ft).
I know it's a lot. im sorry =(
a) radius of a circle
a=pi*r^2
to write this as a function of r
a("r")=pi*r^2
pug in what r is to find what a equals
b) r(t) = 2.2t
where you plug in what your t is to give you r
try using what I gave you for part a and b to find out the rest. Post if you have questions.
The radius of the circle is increasing at a rate of 2.2ft ??? per second, minute, hour? It is kind of important later, as in f) h) i) J) k)
area=pi*r^2
d area/dt= 2 PI r dr/dt
so depending on the answer to my question above, dr/dt is known.
So d area/dt is then known
for j, dr/dt=dl/dt or Time=L/dr/dt_
where L is the distance to shore
the statement <<The radius of the circle is increasing at a rate of 2.2ft. >> is inadequate. Time has to be stated for the 2.2 ft, otherwise it is not a rate.
PLEASE PROOF READ - I assume you mean
2.2ft. / HOUR
a) A = pi r^2
b) 2.2 t
c) 2.2 * 2
2.2 * 2.5
d)A(2) = pi * (4.4)*2
A(2.5) = pi (5.5)^2
e) A(t) = pi (2.2 t)^2
f) A(2) = pi (2.2*2)^2
A(2.5) = pi (2.2*2.5)^2
g) [ A(2.5) -A(2) ]/.5
h) [A(7.)-A(5)]/ 2.5
i) you will see that it increases
by the way
dA/dt = 2 pi r dr/dt
the circumference times how fast the radius is increasing (draw a picture)
j) r = 2.2 t from part b
150 yards r = 3 * 150 = 450 feet
2.2 t = 450
t = 450/2.2
k) , you will have to do this trig yourself. Segment of circle is length 6*580
450^2 + (3*580)^2 = r^2 = (2.2 t)^2
No problem, I can help you with all the parts of your assignment. Let's go step by step to answer each question.
a) To write the area of the circle of oil as a function of the radius, we need to use the formula for the area of a circle, which is A = πr^2. Since the radius is increasing at a rate of 2.2ft, we can express the area of the circle as a function of the radius as follows:
A(r) = πr^2.
b) To write the radius of the circle as a function of time, we need to consider that the radius is increasing at a constant rate of 2.2ft. Therefore, we can express the radius as a linear function of time:
r(t) = 2.2t, where t is the time in hours.
c) To find the radius of the circle of spilled oil after 2 hrs and 2.5 hrs, we can substitute the respective values of t into the equation r(t) = 2.2t.
For 2 hrs: r(2) = 2.2(2) = 4.4 ft.
For 2.5 hrs: r(2.5) = 2.2(2.5) = 5.5 ft.
d) Using the results from part c, we can determine the area of the circle after 2 hrs and 2.5 hrs. We can substitute the values of r into the equation A(r) = πr^2.
For 2 hrs: A(4.4) = π(4.4)^2 ≈ 60.8 sq ft.
For 2.5 hrs: A(5.5) = π(5.5)^2 ≈ 95.0 sq ft.
e) To write the function that represents the area of the circle of spilled oil as a function of time, we can combine the functions from parts a and b. Substituting r(t) into A(r), we get:
A(t) = π(r(t))^2 = π(2.2t)^2 = 4.84πt^2.
f) Using the function from part e, we can determine the area of the circle after 2 hrs and 2.5 hrs by substituting the respective values of t into A(t):
For 2 hrs: A(2) = 4.84π(2)^2 ≈ 30.56π sq ft.
For 2.5 hrs: A(2.5) = 4.84π(2.5)^2 ≈ 60.5π sq ft.
g) To calculate the average rate of change of the area of the circle from 2 hrs to 2.5 hrs, we can use the formula:
Average Rate of Change = (Change in Area) / (Change in Time)
= [(A(2.5) - A(2)) / (2.5 - 2)]
= [(60.5π - 30.56π) / 0.5]
≈ 59.88π sq ft/hr.
h) To calculate the average rate of change of the circle from 5 hrs to 7.5 hrs, we can use the same formula:
Average Rate of Change = (Change in Radius) / (Change in Time)
= [(r(7.5) - r(5)) / (7.5 - 5)]
= [(2.2(7.5) - 2.2(5)) / 2.5]
= [(16.5 - 11) / 2.5]
= 2.2 ft/hr.
i) Based on the results from parts g and h, we can observe that the average rate of change of the area of the circle is increasing with time. In part g, the rate of change is 59.88π sq ft/hr, while in part h, it is 2.2 ft/hr. This indicates that as time passes, the rate of oil spillage and the resulting area of the spilled oil are increasing.
j) To determine when the oil spill will reach the shoreline, we need to find the distance of the shoreline from the oil tanker. You mentioned that the oil tanker is 150 yards from the shore. In order to calculate the time it takes for the spill to reach the shoreline, we need to convert 150 yards to feet: 150 yards * 3 ft/yard = 450 ft. Since the radius of the circle is increasing at a rate of 2.2 ft/hr, we can set up the equation:
r(t) = 2.2t = 450
Solving for t, we find:
t = 450/2.2 ≈ 204.55 hrs.
So, the oil spill will reach the shoreline in approximately 204.55 hrs.
k) To determine how long it will be until 6 miles of shore line is contaminated with oil, we need to convert 6 miles to feet: 6 miles * 5280 ft/mile = 31,680 ft. Since the radius of the circle is increasing at a rate of 2.2 ft/hr, we can set up the equation:
r(t) = 2.2t = 31,680
Solving for t, we find:
t = 31,680/2.2 ≈ 14,400 hrs.
So, it will take approximately 14,400 hrs for 6 miles of shoreline to be contaminated with oil.
I hope this explanation helps you with your assignment! Let me know if you need any further clarification.