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36 problems match “optimization”. Clear

Minimum surface area of a square prism ●●○○○ 20 ptsA closed rectangular prism has a square base with side length $L$ and height $H$ . Its volume is $10000$ . Find the minimum possible surface area. Round to three decimal places.
Minimum sum of squares after error ●●○○○ 20 ptsAshok multiplied three real numbers. He used 73 instead of 37 for one of the numbers, and as a result the product increased by 720. Find the minimum possible value of the sum of the squares of the other two numbers.
Maximum of x+y under a trig constraint ●●○○○ 20 ptsFind the maximum value of $x+y$ if $(2\sin x-1)(2\cos y-\sqrt{3})=0$ , $x\in[0,3\pi/2]$ , $y\in[\pi,2\pi]$ . Round to two decimal places.
Open-top box with minimum surface area ●●○○○ 20 ptsA container company makes an open-top rectangular box with a square base. The box must have a volume of $108\ \text{cm}^3$ . Let the side length of the square base be $x$ cm and the height be $h$ cm. (1) Find the value of $x$ that gives the minimum outer surface area. (2) Find the minimum surface area, in $\text{cm}^2$ .
Minimise |z|² under a conjugate constraint ●●○○○ 20 ptsOver complex numbers $z$ , minimize $z^*z$ subject to $z = z^* + i$ , where $z^*$ is the complex conjugate of $z$ . Write the minimizing $z$ as $z = a + bi$ with real $a$ and $b$ , and find $a$ and $b$ .
Maximum modified distance between unit quaternions ●●○○○ 20 ptsFor unit quaternions $q_1$ and $q_2$ (so $\|q_1\|=\|q_2\|=1$ ), define $d(q_1,q_2)=2\min(\|q_1-q_2\|,\|q_1+q_2\|)$ . Find the maximum value of $d(q_1,q_2)$ . Round to two decimal places.
Max-min difference of a shifted sine ●●○○○ 20 ptsFind the difference between the maximum and minimum values of the function $y=2\sin\left(\frac{\pi x}{6}-\frac{\pi}{3}\right)$ for $0\leqslant x\leqslant 9$ . Round to two decimal places.
Camel banana relay to market ●●●○○ 50 ptsCorey the Camel harvests 3,000 bananas in her grove. The market is 1,000 miles away. Corey walks to the market and eats one banana for each mile she walks. She can carry at most 1,000 bananas at a time, and she may stash bananas along the route and pick them up on later trips. Find the maximum number of bananas that can reach the market. Round to two decimal places.
Smallest n with fewer pencils after a gift ●●●○○ 50 ptsI have two cents and Bill has $n$ cents. Bill wants to buy pencils, which come in two packages: one package costs 6 cents for 7 pencils, and the other costs 10 cents for 12 pencils. Bill can spend all $n$ of his cents on some combination of these packages to get $P$ pencils. If I give my two cents to Bill, he can instead spend all $n+2$ of his cents on some combination of these packages to get fewer than $P$ pencils. Find the smallest value of $n$ for which this is possible.
Smallest entry sum for a conjugating matrix ●●●○○ 50 ptsFor some matrix $\mathbf{P} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with nonzero integer entries, $\begin{pmatrix} 4 & 3 \\ -1 & 0 \end{pmatrix} = \mathbf{P}^{-1} \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \mathbf{P}$ . Find the smallest possible value of $|a| + |b| + |c| + |d|$ .