A circular sector of perimeter 60 m with max | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $r$ be the radius and $	heta$ be the central angle in radians of the circular sector.The perimeter $P$ of the sector is given by $P = 2r + r\

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(B) Let $r$ be the radius and $	heta$ be the central angle in radians of the circular sector.The perimeter $P$ of the sector is given by $P = 2r + r	heta = 60$.From this,we get $	heta = \frac{60 - 2r}{r}$.The area $A$ of the sector is given by $A = \frac{1}{2} r^2 	heta$.Substituting $	heta$,we get $A(r) = \frac{1}{2} r^2 \left( \frac{60 - 2r}{r} ight) = \frac{1}{2} r(60 - 2r) = 30r - r^2$.To find the maximum area,we differentiate $A(r)$ with respect to $r$ and set it to zero:$A'(r) = 30 - 2r = 0$.Solving for $r$,we get $r = 15 \ m$.Since $A''(r) = -2 < 0$,the area is maximum at $r = 15 \ m$.

$A$ circular sector of perimeter $60 \ m$ with maximum area is to be constructed. The radius of the circular arc in meters must be: (in $m$)

Similar Questions

Let $a \in R$ and let $f: R \rightarrow R$ be given by $f(x)=x^5-5x+a$. Then
$(A)$ $f(x)$ has three real roots if $a > 4$
$(B)$ $f(x)$ has only one real root if $a > 4$
$(C)$ $f(x)$ has three real roots if $a < -4$
$(D)$ $f(x)$ has three real roots if $-4 < a < 4$

For the function $f(x) = x^{40} - x^{20}$,find the absolute minimum value in the interval $[0, 1]$.

Find the maximum value of $2x^{3}-24x+107$ in the interval $[1,3]$. Also,find the maximum value of the same function in the interval $[-3,-1]$.

If $x$ and $y$ are two positive numbers such that $x+y=32$,then the minimum value of $x^2+y^2$ is,

If $f(x) = x^2e^{-2x}, x > 0$,then the maximum value of $f(x)$ is ......

Vedclass Test Series

Exam Paper Generator

Online Exam Module