Calculus Questions- Help!!!

[ch]

Hey guys can any one shed some light on how to tackle these questions.
Any help is appreciated.

A) Find the value of a and b so that the tangent to the curve y = ax^2 + bx + 11 at x = 2 has the
equation y = 10x − 1.

B) Cylindrical, cardboard postal tubes are made with the restriction that the sum of the length and the circular circumference are 120 cm. What should the dimensions be for maximum volume?

C)A window frame is in the shape of a semicircle joined to a rectangle. Find the maximum area of a window using 300 cm of framework.

[Ancora_Imparo]

A) Since the tangent to the curve at has equation , the gradient of the curve at must be 10. By substituting into the tangent equation, you also know that is a point on the curve.

Now, you have two bits of information form which you can derive two equations. You can then solve them simultaneously. Full answer below if you're stuck.

Spoiler

When , : ---- Eq. 1

When , :

---- Eq. 2

Eq. 2 - Eq. 1:
Sub into Eq.1: , so

[Ancora_Imparo]

B) You know that for a cylinder .
From the information, you also know that the sum of the circumference and length is 120.
That is:

These type of questions that involve finding the maximum or minimum of some value usually involve eliminating variables until you have only 2 left. This allows you to differentiate and then equate to 0, giving you the values needed for a max/min. In this case, try and eliminate one of or using the equations above.

Spoiler

Rearrange for :
Substitute into volume formula:
Differentiate with respect to :
Equate to 0 for max volume: so (r can't be 0)
Substitute into original equation:

C) Very similar to part B... try this one yourself using a similar method to above.
Hint: Your two variables in this case can be the length and width of the rectangle. The radius of the semicircle will be half of one of these (which ever one you want it to be). Form two equations, one involving the area and another involving the perimeter, which is 300 cm. Rearrange, substitute, differentiate, equate to 0 and solve.

[b^3]

B) " that the sum of the length and the circular circumference are 120 cm", so if we let the length of the tube be , and the radius of the cylinder be then the circumference will be given by . So the sum of the length and circumference will give 120, thus
. Now the volume of the cylinder is given by , in this case will be our length. So we need to maximise , but to do that we need in terms of a single variable, so if we rearrange the first equation we found before for , we can then substitute that into to get in terms of . From there you can take the derivative of with respect to , let it equal to find the value of for which it has maximum volume, which will allow you to find the value of as well.
So really a general method for problems like these are
- Work out what you are trying to maximise/minimise
- Work out what variable the above is in terms of
- See if there is a relationship between those two (or more) variable, and find that relationship
- Substitute this back into the original expression so that you have what you want to maximise in terms of one variable only
- Differentiate this expression with respect to that single variable, let that equal and solve for the value which maximises/minimise the initial expression (careful of endpoints)
- Substitute this back in to find what the question was actually asking for (so in some cases volume, or area, or the lengths for maximum or minimum volume e.t.c)

EDIT: Beaten to the question, but the general method should help you anyways.