Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Calculus

C # 1

An advertisement is to contain 75 sq. cms. of printed area. There is a 2 cm margin at the bottom, 3 cm margin on each side and no margin at the top. Find the dimensions of the smallest possible paper.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Calculus

If the printed width is x, then the printed height must be (75/x), so that the total area is 75.

There is a 3cm margin on each side, so 6 must be added to the width. Similarly, there is a 2cm margin at the bottom, so 2 must be added to the height.

Therefore, the equation that needs to be minimised is (x + 6)(75/x + 2).

Multiplying out of brackets makes this become 75 + 450/x + 2x + 12.

Differentiating gives 2 - 450/x².

The paper is smallest when the differential is equal to 0.

2 - 450/x² = 0
450/x² = 2
450 = 2x²
x² = 225
x = 15cm

That also had a negative answer, but it was discarded because we are dealing with length.

Anyway, using this value with the original equation shows that the smallest possible paper is 21 * 7 = 147cm² big.

---

Why did the vector cross the road?
It wanted to be normal.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

C # 2

The radius of a spherical balloon is increasing at the rate of 5 cm per second when inflated by pumping air. Find the rate of increase of (i) its surface area and (ii) its volume, when the radius is 4 cm.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

C # 3

Differentiate the following with respect to x.

(i) x[sup]x[/sup]       (ii) x[sup]sinx[/sup]

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

C # 4

The volume of a cube is increasing at the rate of 7 cubic centimetres per second. How fast is the surface area increasing when the length of the edge is 12cm?

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

irspow

Member

Registered: 2005-11-24

Posts: 1,055

Re: Calculus

---

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

Excellent, irspow!

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

C # 5

The time of a complete oscillation of a simple pendulum of length l is given by the relation T = 2 π √(l/g) where g is a constant. By what percent should the length be changed in order to correct a loss of 2 minutes per day?

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Calculus

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Calculus

C # 6

If

show that x (dy/dx) = y.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Calculus

On C5, please clarify the two minute variation.
For example, 1.) when a perfect clock strikes midnight, this slow clock reads precisely 23 hr 58 minutes,
or, 2.) when this slow clock reaches midnight, the perfect clock reads 0 hr 02 minutes, which is ever so slightly different.
#1 is slower by factor of (23 58/60)/24, while #2 is slower by factor of 24/(24 2/60).
#1 is less perfect a clock than #2 because it is analogous to 2/3's and 3/4's or  9/10's and 10/11's.

---

igloo myrtilles fourmis

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Calculus

666 bro

Member

From: Flatland

Registered: 2019-04-26

Posts: 706

Re: Calculus

Last edited by 666 bro (2021-04-02 03:03:02)

---

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan