jason9394

jason9394

05.03.2021 •

Computers and Technology

Given an array A[1..n] representing a sequence of n integers, a subsequence is a subset of elements of A, in the same order as they appear in A. A subsequence is monotonic if it is a sequence of strictly increasing numbers. Define LMS(i) to be the length of a longest monotonically increasing subsequence of A[1..i] that must have A[i] as its last element. Write a recurrence for LMS(i) and convert into a dynamic program that calculates LMS(i) for i

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Ответ:

brooklyn4932

11.02.2021

Computers and Technology

$Radius = 1.997\ in$ and $Height = 7.987\ in$

$Cost = \$1.05$

Step-by-step explanation:

Given

Volume = 100in^3

$Cost =\$0.014$ -- Top and Bottom

$Cost =\$0.007$ --- Sides

Required

What dimension of the cylinder minimizes the cost

The volume (V) of a cylinder is:

$V = \pi r^2h$

Substitute 100 for V

$100 = \pi r^2h$

Make h the subject

$h = \frac{100 }{\pi r^2}$

The surface area (A) of a cylinder is:

$A = 2\pi r^2 + 2\pi rh$

Where

$Top\ and\ bottom = 2\pi r^2$

$Sides = 2\pi rh$

So, the cost of the surface area is:

$C = 2\pi r^2 * 0.014+ 2\pi rh * 0.007$

$C = 2\pi r(r * 0.014+ h * 0.007)$

$C = 2\pi r(0.014r+ 0.007h)$

Substitute $h = \frac{100 }{\pi r^2}$

$C = 2\pi r(0.014r+ 0.007*\frac{100 }{\pi r^2})$

$C = 2\pi r(0.014r+ \frac{0.007*100 }{\pi r^2})$

$C = 2\pi r(0.014r+ \frac{0.7}{\pi r^2})$

$C = 2\pi (0.014r^2+ \frac{0.7}{\pi r})$

Open bracket

$C = 2\pi *0.014r^2+ 2\pi *\frac{0.7}{\pi r}$

$C = 0.028\pi *r^2+ \frac{2\pi *0.7}{\pi r}$

$C = 0.028\pi *r^2+ \frac{2 *0.7}{r}$

$C = 0.028\pi *r^2+ \frac{1.4}{r}$

$C = 0.028\pi r^2+ \frac{1.4}{r}$

To minimize, we differentiate C w.r.t r and set the result to 0

$C' = 0.056\pi r - \frac{1.4}{r^2}$

Set to 0

$0 = 0.056\pi r - \frac{1.4}{r^2}$

Collect Like Terms

$0.056\pi r = \frac{1.4}{r^2}$

Cross Multiply

$0.056\pi r *r^2= 1.4$

$0.056\pi r^3= 1.4$

Make r^3 the subject

$r^3= \frac{1.4}{0.056\pi }$

$r^3= \frac{1.4}{0.056 * 3.14}$

$r^3= \frac{1.4}{0.17584}$

r^3= 7.96178343949

Take cube roots of both sides

$r= \sqrt[3]{7.96178343949}$

r= 1.997

Recall that:

$h = \frac{100 }{\pi r^2}$

$h = \frac{100 }{3.14 * 1.997^2}$

$h = \frac{100 }{12.52}$

h = 7.987

Hence, the dimensions that minimizes the cost are:

$Radius = 1.997\ in$ and $Height = 7.987\ in$

To calculate the cost, we have:

$C = 2\pi r(0.014r+ 0.007h)$

C = 2* 3.14 * 1.997 * (0.014*1.997+ 0.007*7.987)

$Cost = \$1.05$

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