Engine displacement is the measure of the cylinder volume

Question:- Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.

The cylinder bore in the form circular cylinder open at the top is to be made from a metal sheet of area 75 π cm².

Based on the above information, answer the following questions:

(i) If the circular cylinder is ‘r’ cm and height is ‘h’ cm, then write the volume V of the cylinder in term of radius ‘r’.

(ii) Find $\frac{dV}{dr}$ .

(iii) (a) Find radius of cylinder when its volume is maximum.

(b) For maximum volume, h > r . State true or false and justify.

Solution :–

(i) The cylinder bore in the form circular cylinder open at the top is to be made from a metal sheet of area = $75\pi$

$2\pi r h +\pi r^2 = 75 \pi$

Divide by $\pi$

$\Rightarrow 2 r h + r^2 = 75$

$\Rightarrow 2 r h = 75-r^2$

$\Rightarrow h =\dfrac{75-r^2}{2r}---(1)$

Volume of cylinder $= \pi r^2 h$

$V = \pi r^2 \left(\dfrac{75-r^2}{2r}\right)$

$\Rightarrow V = \frac{1}{2}\pi r(75-r^2)$

$\Rightarrow V =\frac{1}{2}\pi(75r-r^3)$

(ii) Since, $V =\frac{1}{2}\pi(75r-r^3)---(2)$

Differentiate with respect to r

$\dfrac{dV}{dr}=\dfrac{1}{2}\pi (75-3r^2)----(3)$

(iii) For maxima and minima $\dfrac{dV}{dr} =0$

Hence, $\dfrac{1}{2}(75-3r^2)=0$

$\Rightarrow r^2 = \dfrac{75}{3}=25$

$\Rightarrow r = 5$

Again differentiate with respect to r of eq (3)

$\dfrac{d^2V}{dr^2}= \dfrac{1}{2}(-6r)=-3r<0$

Volume of cylinder is maximum when $r = 5$

(b) Value of $r = 5$ putting in eq (1)

$h =\dfrac{75-r^2}{2r}$

$\Rightarrow h = \dfrac{75-(5)^2}{2(5)}$

$\Rightarrow h = \dfrac{75-25}{10}$

$\Rightarrow h = \dfrac{50}{10}$

$\Rightarrow h = 5$

Hence $h = r$

For maximum volume, h > r is false statement

Question:- The use of electric vehicle will curb air pollution in the long run.

The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time t is given by the function V:

$V(t) =\frac{1}{5}t^3-\frac{5}{2}t^2 + 25 t -2$

Where t represents the time and t = 1, 2, 3 . . . corresponds to year 2001,2002, 2003 …. respectively.

Based on the above information, answer the following question:

(i) Can the above function be used to estimate number of vehicles in the year 2000 ? Justify.

(ii) Prove that the function V(t) is an increasing function.

Solution:- For solution click here

Question :- Recent studies suggest that roughly 12 % of the world population is left handed.

Depending upon the parents, the chances of having a left handed child are as follows :

A : When both father and mother are left handed :

Chances of left handed child is 24 %

B : When father is right handed and mother is left handed :

Chances of left handed child is 22 %

C : When father is left handed and mother is right handed :

Chances of left handed child is 17 %

D : When both father and mother are right handed :

Chances of left handed child is 9 %

Assume that $P(A) = P(B) = P(C) = P(D) = \frac{1}{4}$ and L denotes the event that child is left haanded.

Based on the above information, answer the following question :

(i) Find $P(L/C)$

(ii) Find $P(\overline{L}/A)$

(iii) (a) Find $P(A/L)$

(b) Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.

Solution :-For solution click here

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