A triangular garden is fenced on two sides,an | vedclass

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

Question

(B) Let the two equal sides of the triangle be $x$ and the angle between them be $2	heta$. From the geometry of the triangle,the height is $h = x \si

Vedclass · Accepted Answer

$\frac{1}{2}x^2$

Vedclass · Answer

(B) Let the two equal sides of the triangle be $x$ and the angle between them be $2	heta$. From the geometry of the triangle,the height is $h = x \sin 	heta$ and the base is $b = 2x \cos 	heta$. The area of the triangle is given by $A = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height}$. $A = \frac{1}{2} 	imes (2x \cos 	heta) 	imes (x \sin 	heta) = x^2 \sin 	heta \cos 	heta$. Using the identity $\sin 2	heta = 2 \sin 	heta \cos 	heta$,we get $A = \frac{1}{2} x^2 \sin 2	heta$. To maximize the area $A$,we need to maximize $\sin 2	heta$. The maximum value of $\sin 2	heta$ is $1$,which occurs when $2	heta = 90^\circ$ or $	heta = 45^\circ$. Therefore,the maximum area is $A_{\max} = \frac{1}{2} x^2 (1) = \frac{1}{2} x^2$.

$A$ triangular garden is fenced on two sides,and the third side is a straight river bank. The two fenced sides have an equal length of $x$. What is the maximum area that can be enclosed by the garden?

Similar Questions

If the function $f(x)=2x^3-9ax^2+12a^2x+1, a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$,then $\alpha$ and $\alpha^2$ are the roots of the equation:

If $x$ is real,then the greatest and least values of $\frac{x^2 - x + 1}{x^2 + x + 1}$ are

If the function $f(x) = \frac{t + 3x - x^2}{x - 4}$,where $t$ is a parameter,has a local maximum and a local minimum,then the range of values of $t$ is:

The equations of motion of two stones thrown vertically upwards simultaneously are $s_1 = 19.6t - 4.9t^2$ and $s_2 = 9.8t - 4.9t^2$ respectively. If the maximum height attained by the first stone is $h$,then when the first stone is at its maximum height,the height of the second stone will be:

The range of $a \in R$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ where $x \neq 2 n \pi, n \in N$,has critical points,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module