Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applied max/min questions.
Type 4: Advanced 3D shapes - Test Yourself 1 - Solutions.


 

Box in a sphere. 1.

(i)

(ii)

(iii)

To test turning point:

x 7 7.3 8
dV/dx 73.1 0 -286

So a change of gradient from +ve to -ve - ∴ max value.
Max vol is 1232 cm3.
  2.
Cone in a sphere.

3. (i) The volume of the cone would be . Looking at the required equation in the question, there is no r pronumeral. So this has to be eliminated in the equation here. Draw the following picture to create another triangle - not an uncommon strategy (so remember it). Then use Pythagoras.

(ii) Expand immediately so we do not have to use a product rule.
Also recognise that R is the radius of the sphere - AND SO IT IS A CONSTANT!!
 

4. (i)

(ii)

(iii)

   
   
Cylinder in a sphere. 5.
 

6.

 

 
Cylinder in a cone.

7.

 

(i)

(ii)

(iii)

 

8.