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The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
Concept: undefined >> undefined
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Concept: undefined >> undefined
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Evaluate the following.
`intx^3 e^(x^2) dx`
Concept: undefined >> undefined
Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.
Concept: undefined >> undefined
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Concept: undefined >> undefined
Three new machines M1, M2, M3 are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M2 can not be placed at B. The cost matrix (in hundred rupees) is as follows:
| Machines | Places | |||
| A | B | C | D | |
| M1 | 13 | 10 | 12 | 11 |
| M2 | 15 | - | 13 | 20 |
| M3 | 5 | 7 | 10 | 6 |
Determine the optimum assignment schedule and find the minimum cost.
Concept: undefined >> undefined
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
Concept: undefined >> undefined
Evaluate the following:
`intx^3e^(x^2)dx`
Concept: undefined >> undefined
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Concept: undefined >> undefined
Evaluate the following.
`intx^3 e^(x^2) dx`
Concept: undefined >> undefined
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Concept: undefined >> undefined
Evaluate the following.
`intx^3e^(x^2) dx`
Concept: undefined >> undefined
Evaluate `int (1 + x + x^2/(2!))dx`
Concept: undefined >> undefined
Choose the correct pair:
| Group A | Group B |
| 1) Price Index | a) `(sump_1q_1)/(sump_0q_0)xx100` |
| 2) Value Index | b) `(sumq_1)/(sumq_0)xx100` |
| 3) Quantity Index | c) `(sump_1q_1)/(sump_0q_1)xx100` |
| 4) Paasche's Index | d) `(sump_1)/(sump_0)xx100` |
Concept: undefined >> undefined
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Concept: undefined >> undefined
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Concept: undefined >> undefined
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Concept: undefined >> undefined
Evaluate the following.
`int x^3 e^(x^2) dx`
Concept: undefined >> undefined
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Concept: undefined >> undefined
Evaluate the following.
`intx^2e^(4x)dx`
Concept: undefined >> undefined
