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36 problems match “optimization”. Clear

Stiffest rectangular beam from a log ●●●○○ 50 ptsThe stiffness of a beam is proportional to its width times the cube of its depth. Find the width and depth of the stiffest rectangular beam that can be cut from a cylindrical log of diameter 12 inches. Give the width and depth in inches. Round to three decimal places.
Maximum traffic flow over average speed ●●●○○ 50 ptsOn a certain road section, the traffic flow $y$ (in thousands of vehicles per hour) and the average speed $v$ (in km/h) of vehicles satisfy $y=\frac{670(v-5)}{v^2-8v+915}$ for $v>5$ . Find the average speed $v$ at which the traffic flow is largest, and the maximum traffic flow. Round the maximum traffic flow to one decimal place.
Largest finite count of small elements ●●●●○ 100 pts $S$ is a set of complex numbers such that if $u, v \in S$ , then $uv \in S$ and $u^{2}+v^{2} \in S$ . Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$ ?
Minimum of a sum of |1+z^k| ●●●●○ 100 ptsLet the complex number $z$ satisfy $|z|=1$ . Find the minimum possible value of $2021|1+z|+|1+z^{2}|+|1+z^{3}|+\cdots+|1+z^{4043}|$ .
Minimum ratio of derivative energy to squared mean ●●●●○ 100 ptsLet $f$ be a continuously differentiable real-valued function on $[0,1]$ with $\int_{1/3}^{2/3} f(x)\,dx = 0$ . Find the minimum value of $\dfrac{\int_0^1 (f'(x))^2\,dx}{\left(\int_0^1 f(x)\,dx\right)^2}$ over all such $f$ for which the denominator is nonzero.
Maximum of |bc| for a bounded quadratic ●●●●● 200 ptsLet complex numbers $a$ , $b$ , $c$ satisfy: for any complex number $z$ with $|z|\leq 1$ , we have $|az^2+bz+c|\leq 1$ . Find the maximum value of $|bc|$ . Round to two decimal places.