Download App

Maxima and Minima — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima5 Marks
Question
Find the largest possible area of a right angled triangle whose hypotenuse is 5cm long.
✓

Answer

Let the base of right angled triangle be x and its height be y, Then,
$\text{x}^{2}+\text{y}^{2}=5^{2}$
$\Rightarrow\text{y}^{2}=25-\text{x}^{2}$
$\Rightarrow\text{y}=\sqrt{25-\text{x}^{2}}$
As, thr area of the triangle, $\text{A}=\frac{1}{2}\times\text{x}\times\text{y}$
$\Rightarrow\text{A}(\text{x})=\frac{1}{2}\times\text{x}\times\sqrt{25-\text{x}^{2}}$
$\Rightarrow\text{A}(\text{x})=\frac{\text{x}\sqrt{25-\text{x}^{2}}}{2}$
$\Rightarrow\text{A}'(\text{x})=\frac{\sqrt{25-\text{x}^{2}}}{2}+\frac{\text{x}(-2\text{x})}{4\sqrt{25-\text{x}^{2}}}$
$\Rightarrow\text{A}'(\text{x})=\frac{\sqrt{25-\text{x}^{2}}}{2}+\frac{\text{x}^{2}}{2\sqrt{25-\text{x}^{2}}}$
$\Rightarrow\text{A}'(\text{x})=\frac{25-\text{x}^{2}-\text{x}^{2}}{2\sqrt{25-\text{x}^{2}}}$
$\Rightarrow\text{A}'(\text{x})=\frac{25-2\text{x}^{2}}{2\sqrt{25-\text{x}^{2}}}$
For maxima or minima, We must have f'(x) = 0
A'(x) = 0
$\Rightarrow\frac{25-2\text{x}^{2}}{2\sqrt{25-\text{x}^{2}}}=0$
$\Rightarrow 25-2\text{x}^{2}=0$
$\Rightarrow 2\text{x}^{2}=25$
$\Rightarrow \text{x}=\frac{5}{\sqrt{2}}$
So, $\text{y}=\sqrt{25-\frac{25}{2}}$
$=\sqrt{\frac{50-25}{2}}$
$=\sqrt{\frac{25}{2}}$
$={\frac{5}{\sqrt{2}}}$
Also, $\text{A}'(\text{x})=\frac{\Bigg[-4\text{x}\sqrt{25-\text{x}^{2}}-\frac{(25-2\text{x}^{2})(-2\text{x})}{2\sqrt{25-\text{x}^{2}}}\Bigg]}{25-\text{x}^{2}}$
$=\frac{\Bigg[\frac{-4\text{x}(25-\text{x}^{2})+(25\text{x}-2\text{x}^{3})}{\sqrt{25-\text{x}^{2}}}\Bigg]}{25-\text{x}^{2}}$
$=\frac{-100\text{x}+4\text{x}^{3}+25\text{x}-2\text{x}^{3}}{(25-\text{x}^{2})\sqrt{25-\text{x}^{2}}}$
$=\frac{-75\text{x}+2\text{x}^{3}}{(25-\text{x}^{3})\sqrt{25-\text{x}^{2}}}$
$\Rightarrow\text{A}''\Big(\frac{5}{\sqrt{2}}\Big)=\frac{-75\Big(\frac{5}{\sqrt{2}}\Big)+2\Big(\frac{5}{\sqrt{2}}\Big)^{3}}{\Bigg(25-\Big(\frac{5}{\sqrt{2}}\Big)^{2}\Bigg)^\frac{3}{2}}<0$
So, $\text{x}=\Big(\frac{5}{\sqrt{2}}\Big)$ is point of maxima.
$\therefore$ The largest possible area of triangle $=\frac{1}{2}\times\Big(\frac{5}{\sqrt{2}}\Big)\times\Big(\frac{5}{\sqrt{2}}\Big)=\frac{25}{4}$ square units.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Maxima and Minima→Maxima and Minima — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$
→
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi:(\pi+2)$.
→
If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that $\overrightarrow{\text{OA}}+\overrightarrow{\text{OB}}+\overrightarrow{\text{OC}}=\overrightarrow{\text{OD}}+\overrightarrow{\text{OE}}+\overrightarrow{\text{OF}}$.
→
If $\text{y}=\frac{\text{x}}{\text{x}+2},$ show that $\text{x}\frac{\text{dy}}{\text{dx}}=(1-\text{y})\text{y}$
→
Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in
  1. Exactly one college.
  2. At most two colleges.
  3. At least two colleges.
→
Evaluate the following definite integrals:$\int_{1}^\limits{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$
→
Solve the following system of equations by matrix method:
$3x + y = 19$
$3x - y = 23$
→
$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$
→
Show that the following system of linear equations is consistent and also find solution:
$6x + 4y = 2$
$9x + 6y =3$
→
Let S be the set of all real numbers except -1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
→