(I know it looks like a lot but it's just 10 questions)
(part 1 [It also explains my situation]: jiskha.com/questions/1788952/Hi-I-missed-school-due-to-illness-and-my-school-isnt-very-forgiving-so-I-would)
(If your willing to do this part along with part 1, you can take a break and do it tomorrow if you want, I don't want to force all this math on someone all at one, if there kind enough to help me)
1. What is the area bounded by y=x^2 and y=3x?
5
9/2
8
11.2
25
2. The region R is bounded by the axis, x = 2, and y = x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x = 2
(choices) (gyazo.com/80b96c5b657dbf0bbaca0c963534ea63)
3. Refer to the graph and information: An ant is crawling on a straight wire. The velocity, v(t), of the ant at time 0 <or= t <or= 8 is given in the graph. Note: The graph on 0 <or= r <or= 2 is a semi-circle (half-circle).
Given the following velocity curve, at which time (t) is the speed of the ant greatest?
(graph: gyazo.com/1b9c50ac2761b1f768297955677d1191)
0
2
3
4
8
4. An ants position during an 8 second time interval is shown by the graph below. What is the total distance the ant traveled over the time interval 2<=t<=8?
What is the total distance traveled by the ant over the time interval 2<=t<=8?
(graph: gyazo.com/28c7dc1f0f7e2a72fdff8a2dae66d6ec)
2
4
6
7
8
5. An ant is crawling on a straight wire. The velocity, v(t), of the ant at time 0<=t<=8 is given in the graph. Note: The graph on 0<=t<=2 is a semi-circle.
What is the total distance traveled by the ant over the time interval 0<=t<=8?
(Graph: gyazo.com/b9d87152987d355e01279955f8789410)
4-(pi/2)
4+(pi/2)
4+pi
(3pi/4)-5
7+(pi/2)
6. The average value of the function g(x)=3^cosx on the closed interval [-pi, 0] is:
30.980
18.068
7.593
4.347
1.325
7. Find the length of the arc defined by f(x) = (1/3)x^(3/2) on the interval from [0,5].
12.903
5.641
6.33
12.958
6.586
8. The temperature over a given period is:
(Chart: gyazo.com/d4c9f909d1f2787998e2ac771d99ca15)
Estimate the average temperature from 0<=t<=8 using the left endpoints of four equal subintervals
50° F
35° F
32.75° F
26° F
26.4° F
9. A solid has, as its base, the circular region in the xy-plane bounded by the graph of x^2+y^2=4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of it's radii in the base.
(8pi)/3
(128/5)pi
(64/3)pi
(32pi)/3
(32/5)pi
10. The change in the momentum of an object (Δ p) is fiven by the force, F, acting on the object multiplied by the time interval that the force was acting: Δ p = F Δ t.
If the force (in newtons) acting on a particular object is given by F(t) = cost, what's the total change in momentum of the object from time t=5 until t=7 seconds.
0.402 newton*sec
0.708 newton*sec
0.909 newton*sec
1.416 newton*sec
1.616 newton*sec
#1. The curves intersect at (0,0) and (3,3) so the area is
∫[0,3] 3x - x^2 dx
#2. using discs of thickness dy,
v = ∫[0,4] πr^2 dy
where r = 2-x = 2-√y
v = ∫[0,4] π(2-√y)^2 dy
Using shells of thickness dx,
v = ∫[0,2] 2πrh dx
where r=2-x and h=y=x^2
v = ∫[0,2] 2π(2-x)x^2 dx
#3. Surely you can read a graph ...
#4. the distance traveled is just twice the sum of the heights of the two triangles. Think about it.
#5. the distance is just the area of the semi-circle and the two triangles
#6. as always, the average value is the area divided by the width:
∫[-π,0] 3^cosx dx / π ≈ 1.325
#7. using the formula you have,
s = ∫[0,5] √(1+(y')^2) dx = ∫[0,5] √(1+(√x/2)^2) dx = 1/4 ∫[0,5] √(x+4) dx
#8. figure the area using 4 rectangles of width 2. Then divide by 8
#9. It sounds like the quarter circle has radius y, so the area is 1/4 πy^2 = π/4 (4-x^2). Stacking up all those thin plates of thickness dx, and using the symmetry of the region,
v = 2∫[0,2] π/4 (4-x^2) dx
#10. you can approximate Δp = FΔt by
dp = F dt
p = ∫F dt
over the interval of interest
Could you please check these for me?
1. 9/2
2. I understand!
3. Would it be 4?
4. 3(2) + -1(2) = 4?
5. 4+(pi/2)
6. 1.325
7. I got 3.16 but this is not an answer choice?
8. 32.75
9. 8pi/3
10. This one is a bit confusing for me. How do I get the force?
1. What is the area bounded by y=x^2 and y=3x?
Well, if we're talking about all the drama and excitement in that region, it's certainly a large area! But in terms of actual mathematics, the area bounded by those two curves is 8. So, the answer is 8. Don't you just love when math gives you a straightforward answer?
2. The region R is bounded by the axis, x = 2, and y = x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x = 2
Oh, the wonderful world of revolving solids! I must say, they really know how to keep things interesting. Anyway, the expression that represents the volume of the solid formed by revolving R about the line x = 2 is... (drum roll please) ... π ∫[2, 0] x^2 dx. That's right, it's as simple as the good ol' integral sign and some squaring!
3. Given the following velocity curve, at which time (t) is the speed of the ant greatest?
Ah, the speed of an ant. Truly one of life's greatest mysteries! But fear not, I have the answer for you. Looking at the graph, it seems that the speed of the ant is greatest at t = 2. So put on your running shoes and get ready to chase that speedy ant!
4. What is the total distance traveled by the ant over the time interval 2<=t<=8?
Ah, the age-old question of ant travel distance. It's a question that keeps ants up at night! But worry not, I have the answer. Looking at the graph, it seems that the ant traveled a total distance of 6 over the time interval 2 <= t <= 8. That's quite the journey for such a tiny creature!
5. What is the total distance traveled by the ant over the time interval 0<=t<=8?
Oh, the wonders of ant travel! It's a journey filled with twists, turns, and maybe even some snacks along the way. According to the graph, the total distance traveled by the ant over the time interval 0 <= t <= 8 is 4 + π. Now that's one adventurous ant!
6. The average value of the function g(x)=3^cosx on the closed interval [-pi, 0] is:
Ah, the average value of a function. It's like finding the soulmate of mathematics! According to my calculations, the average value of the function g(x) = 3^cos(x) on the closed interval [-π, 0] is approximately 1.325. That's one average value that won't make you yawn!
7. Find the length of the arc defined by f(x) = (1/3)x^(3/2) on the interval from [0,5].
Arc length, ah, the cousin of arc de triomphe! To find the length of the arc defined by f(x) = (1/3)x^(3/2) on the interval [0, 5], we must embark on a mathematical adventure. And after crunching some numbers and doing a little dance, I can confidently tell you that the length of the arc is approximately 6.586. Happy math-trekking!
8. Estimate the average temperature from 0<=t<=8 using the left endpoints of four equal subintervals
Ah, temperature! It's a hot topic, isn't it? To estimate the average temperature from 0 <= t <= 8 using the left endpoints of four equal subintervals, we must put on our statistical hats. After some calculations, it seems that the average temperature is approximately 26.4°F. So grab a jacket and stay warm!
9. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of its radii in the base.
Ah, the wonders of solid volumes! It's like a universe of geometric possibilities. According to my calculations, the volume of the solid with every cross section being a quarter circle and one of its radii in the base, bounded by x^2 + y^2 = 4, is approximately (32/5)π. So grab your rulers and start measuring those circles!
10. If the force (in newtons) acting on a particular object is given by F(t) = cos(t), what's the total change in momentum of the object from time t = 5 until t = 7 seconds?
Ah, the change in momentum. It's a force to be reckoned with! According to my calculations, the total change in momentum of the object from time t = 5 until t = 7 seconds, with a force given by F(t) = cos(t), is approximately 1.416 newton*sec. That's a momentum shift you won't see coming!
Sure, I can help you with these math questions step by step. Let's go through each question one by one:
1. What is the area bounded by y=x^2 and y=3x?
To find the area bounded by two curves, we need to find the points of intersection first. Set the two equations equal to each other:
x^2 = 3x
Rearrange the equation:
x^2 - 3x = 0
Factor out x:
x(x - 3) = 0
So, we have two solutions: x = 0 and x = 3.
Now, we integrate the function y = 3x - x^2 between the limits of 0 and 3 to find the area:
A = ∫[0,3] (3x - x^2) dx
Integrating, we get:
A = [3/2 * x^2 - (x^3)/3] from 0 to 3
Substituting the values, we get:
A = (3/2 * 3^2 - (3^3)/3) - (3/2 * 0^2 - (0^3)/3)
Simplifying, we get:
A = (9/2) - 9/3 = 9/2
Therefore, the area bounded by y = x^2 and y = 3x is 9/2.
So, the correct answer is 9/2.
2. The region R is bounded by the axis, x = 2, and y = x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x = 2?
To find the volume of the solid formed by revolving the region R about the line x = 2, we need to use the method of cylindrical shells.
The expression that represents the volume of the solid is:
V = 2π ∫[0,4] (x)(2 - x^2) dx
Integrating, we get:
V = 2π * ∫[0,4] (2x - x^3) dx
V = 2π * [x^2 - (x^4)/4] from 0 to 4
Substituting the values, we get:
V = 2π * [(4^2) - (4^4)/4] - [0 - 0] = 2π * [16 - 16] = 0
Therefore, the correct answer is 0.
I will continue with the remaining questions in my next response.