# Major Calculus Assignment Issue

• October 31st 2006, 06:13 PM
philipsach1
Major Calculus Assignment Issue
I know I put one of the questions up earlier, but it would really help me out if someone could give me some input on the rest of these.

1. (a) Determine the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of length 5 cm and 12 cm. The two sides of the rectangle lie along the legs.
(b) Repeat part (a) with a right triangle that has sides 8 cm by 15 cm.
(c) Hypothesize a conclusion for any right triangle.

2. Determine the dimensions of the cone of maximum volume that can be inscribed in a sphere with radius 3?

3. The position, s, of an object moving in a straight line from a fixed point at t seconds is given by
s(t)=t^3 - 9t^2 - 21t - 11, where t>0 and s(t) is measured in metres.
(a) Determine when the object is stopped and when it is moving forward.
(b) Is the acceleration negative at any time?
(c) When is the object speeding up and when is the object slowing down? Explain why.

Any help would do wonders.
• October 31st 2006, 06:55 PM
ThePerfectHacker
Quote:

Originally Posted by philipsach1

3. The position, s, of an object moving in a straight line from a fixed point at t seconds is given by
s(t)=t^3 - 9t^2 - 21t - 11, where t>0 and s(t) is measured in metres.

You have, for $t>0$
$s(t)=t^3-9t^2-21t-11$
Thus,
$s'(t)=v(t)=3t^2-18t-21$
Thus,
$s''(t)=v'(t)'=a(t)=6t-18$

Quote:

(a) Determine when the object is stopped and when it is moving forward.
The object is at rest when the speed is zero.
Thus,
$3t^2-18t-21$
Divide by three,
$t^2-6t-7=0$
$(t-7)(t+1)=0$
$t=7,-1$
But time is measure positive here,
$t=7$.

Quote:

(b) Is the acceleration negative at any time?
We need that,
$6t-18<0$
Thus,
$6t<18$
Thus,
$t<3$

Quote:

(c) When is the object speeding up and when is the object slowing down? Explain why.
Look at (b) the object slowing down when acceleration is zero (decelleration) so it is slowing down until 3 seconds.
• October 31st 2006, 07:12 PM
Quick
I hope this helps you get a visual of the first problem...

Move the red points and see when the area is the greatest :)

Tell me if this helps
• October 31st 2006, 09:02 PM
Soroban
Hello, Philip!

Quote:

2. Determine the dimensions of the cone of maximum volume
that can be inscribed in a sphere with radius 3

Code:

                A               * * *           *    /:\    *         *    / : \    *       *    /  :3 \    *             /  :  \       *    /    :    \    *       *  /    O*    \  *       *  /  *  :  * 3 \  *         / *    y:    * \     B *--------+--------* C         *          x  *           *          *               * * *

The center of the sphere is $O.$
Note that $OA = OB = OC = 3.$
$\Delta ABC$ is the side view of the cone.

The volume of a cone is: . $V \:=\:\frac{1}{3}\pi r^2h$

In the diagram, the radius is $x$ and the height is $y + 3$
So we have: . $V \:=\:\frac{1}{3}\pi x^2(y+3)$ [1]

The diagram also says: . $x^2 + y^2 \:=\:3^2\quad\Rightarrow\quad x^2 \:=\:9 - y^2$ [2]

Substitute [2] into [1]: . $V \;= \;\frac{1}{\pi}(9 - y^2)(y + 3)$

We have: . $V \;=\;\frac{\pi}{3}\left(-y^3 - 3y^2 + 9y + 27\right)$

Then: . $V' \;=\;\frac{\pi}{3}\left(-3y^2 - 6y + 9\right) \:=\:0$

We have the quadratic: . $y^2 + 2y = 3 \:=\:0$

. . which factors: . $(y - 1)(y + 3)\:=\:0$

. . and has roots: . $y = 1,\,-3$

Since the measurements are positive, $y = 1$
. . and from [2]: . $x^2 \:=\:9-1^2\:=\:8\quad\Rightarrow\quad x \,= \,2\sqrt{2}$

Therefore, the cone of maximum volume has radius $2\sqrt{2}$ and height $4.$

• November 1st 2006, 05:54 PM
philipsach1
Thanks!
Thank you all so very much. All of these were of great help to me. :>
• November 1st 2006, 06:13 PM
Quick
Quote:

Originally Posted by philipsach1
Thank you all so very much. All of these were of great help to me. :>

I'm just curious (since I'm experimenting with the drawing program) did my diagram help at all? please be honest
• November 1st 2006, 06:15 PM
philipsach1
Re:
It really was helpful. It helped me to visualize the problem. But since I had to look for dimensions, it would have been better if those had been on there, but it was still very helpful. Thank you again.