MATHS CALCULATION

QUESTION

1. Given that,

Therefore, we have,

(a)  When, n=10

Therefore, by trapezoidal rule,

(b) By  Simpson’s rule,

(c)  Exact value of F can be found as

(d) We can see that the result obtained from Simpson’s rule is closer to the result obtained using integration as compared to the result obtained from trapezoidal rule.

1. (a) Given that,

Initially,

Therefore, we have

Therefore, we have

…… (1)

(b) If,

We have, from equation (1),

Therefore, 0.277 seconds have elapsed since then.

(c)

We have, from equation (1),

Therefore, 0.0205 seconds have elapsed since then.

1. The height of the cask

We need to find the equation of a parabola with vertex at (0, 80) and passing through (125, 60)

The equation of side of parabola will be

The parabola passes through (125, 60). So,

So, equation of side of cask is

We need to find the volume of the cask which is generated when we rotate the parabola between -125 and 125 around the x-axis.

SOLUTION

1. Given that,

We substitute these values in the expression for F.

Therefore, we have,

(a)  We have to find the value of F by Trapezoidal rule.

In Trapezoidal rule, we divide the interval [2, 12] into 10 sub –intervals of equal length.

We approximate the integral by using 10 trapezoids formed between

By adding the area of the 10 trapezoids, we obtain

When, n=10

Therefore, we divide the given interval into 10 sub-intervals of equal length 1:

The points can be calculated as shown:

Therefore, by trapezoidal rule,

This means the area of all 10 trapezoids that is area under the curve is 88954.64. Thus, from trapezoidal rule, the value of definite integral is 88954.64.

(b) We have to find the value of F by Simpson’s rule with n=10

Simpson’s rule evaluates a definite integral by using quadratic polynomials. The function is divided into 10 sub intervals, that is even number of subintervals of equal length. We estimate the integral by adding the area under the parabolic arcs through three successive points.

We know that by Simpson’s rule,

We use the formula to evaluate the definite integral F.

Therefore, by Simpson’s rule we get that the area under the curve, that is the definite integral is 88726.21067.

(c)  We have to evaluate exact value of F using integration. We know that

We use the properties of integration to obtain the exact value of F.

Exact value of F can be found as

We can integrate it by parts.

Therefore, we have obtained the exact value of F which is 88754.75.

(d) We have to comment on the results of (a), (b), and (c).

From Trapezoidal rule, we have obtained,

From Simpson’s rule, we have obtained,

And we obtained the exact value by integration to be

Therefore, we can see that the result obtained from Simpson’s rule is closer to the result obtained using integration as compared to the result obtained from trapezoidal rule. Therefore, Simpson’s rule gives more accurate result as compared to Trapezoidal rule.

1. (a) We are Given that,

We have to show that at any time t,

Therefore, we are given a differential equation. We have to find its solution with initial condition that at time t=0, i=i₀.

Therefore, first we shall solve the differential equation by using the method of variable- separable and then under the given initial condition, we shall find its particular solution.

We have,

Initially,

Therefore, we have

Therefore, we have

…… (1)

Therefore, we obtained that any time t,

This means the current through a R-L circuit decays exponentially with the passage of time. It decays by a factor of of the initial value.

(b) We have to find the time elapsed if the current is half of its initial value when the resistance is 5 ohms and inductance is 5 Henry. We have already shown from equation (1) that

We are given,

We shall find the time t, from equation (1),

Therefore, it will take 0.277 seconds before the current reaches half of its initial value. As we already know, the current is decaying exponentially.

(c) We have to find the time elapsed if the current is 95% of its initial value when the resistance is 5 ohms and inductance is 5 Henry. We have already shown from equation (1) that

We are given,

We have to find the time t.

We have, from equation (1),

Therefore, it will take 0.0205 seconds before the current reaches 95% of its initial value. Since the current is decaying exponentially, the decay is very fast.

1. We are given a cask with sides parabolic with radii of the sides to be 60 cm and middle radius 80 cm. The height of the cask is 2.5m.

Therefore, The height of the cask

We lay the cask on its side to interpret the problem.

The shape of the side of cask is a parabola.

Since the total height of the cask is 250 cm and the radii on the sides are 60 cm and radius in the middle is 80 cm, the vertex will be (0,80) and coordinates of two points are (125,60) and (-125,60).

We need to find the equation of a parabola with vertex at (0, 80) and passing through (125, 60)

The equation of side of parabola will be

The parabola passes through (125, 60). So,

So, equation of side of cask is

We need to find the volume of the cask which is generated when we rotate the parabola between -125 and 125 around the x-axis.

WE have used the volume of solid of revolution. By rotating the parabola about the x-axis between -125 and 125, the cask is generated.

Therefore, we get that the capacity of the cask is 4251.622 L

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