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Fudgin [204]
3 years ago
11

A homogeneous rectangular lamina has constant area density ρ. Find the moment of inertia of the lamina about one corner

Mathematics
1 answer:
frozen [14]3 years ago
7 0

Answer:

I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

Step-by-step explanation:

By applying the concept of calculus;

the moment of inertia of the lamina about one corner I_{corner} is:

I_{corner} = \int\limits \int\limits_R (x^2+y^2)  \rho d A \\ \\ I_{corner} = \int\limits^a_0\int\limits^b_0 \rho(x^2+y^2) dy dx

where :

(a and b are the length and the breath of the rectangle respectively )

I_{corner} =  \rho \int\limits^a_0 {x^2y}+ \frac{y^3}{3} |^ {^ b}_{_0} \, dx

I_{corner} =  \rho \int\limits^a_0 (bx^2 + \frac{b^3}{3})dx

I_{corner} =  \rho [\frac{bx^3}{3}+ \frac{b^3x}{3}]^ {^ a} _{_0}

I_{corner} =  \rho [\frac{a^3b}{3}+ \frac{ab^3}{3}]

I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

Thus; the moment of inertia of the lamina about one corner is I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

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7nadin3 [17]

Answer:

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Step-by-step explanation:

The area of a trapezoid is given by

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We know A b1 and b2

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3 years ago
Suppose you had D dollars in your bank account. you spent $22 but have at least $28 left. how much money did you have initially?
nalin [4]

Answer:

Answer:

d--22→ 28

d→ 50

Step-by-step explanation:

Let d dollars be the initial amount in your bank account.

We have been given that you spent $22, so the amount left after spending $22 will be: .

We are also told that after spending $22, you have at least $28. This means that the amount left after spending $22 will be greater than or equal to $28.

We can represent this information in an equation as:

Therefore, the inequality represents the initial amount of money you had.

Now let us solve for d by adding 22 to both sides of our inequality.

Therefore, initially you had at least $50.

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Step-by-step explanation:

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8 0
2 years ago
What is the side length, in inches, of the pets
tatiyna

Candy draws a square design with a side length of x inches for the window at the pet shop. She takes the design to the printer and asks for a sign that has an area of 16x2 – 40x + 25 square inches. What is the side length, in inches, of the pet shop sign?

Answer:

the length of the sign is 4x-5 inches

Step-by-step explanation:

Given

Area of the square of design = 16x^{2} -40x+25

First we find the roots of equation 16x^{2} -40x+25=0

The roots of the quadratic equation ax^{2} +bx^{2} +c=0 are given by

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where a=16, b=-40, c=25

x=\frac{40\pm\sqrt{(-40)^2-4\times 16\times 25}}{2\times 16}

x=\frac{40\pm\sqrt{1600-1600}}{32}

x=\frac{40\pm\sqrt{0}}{32}

x=\frac{40}{32}

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That is, the factors of the polynomial 16x^{2} -40x+25 are 4x-5 and 4x-5.

So, Area of the square design = 16x^{2} -40x+25 = (4x-5)^{2}

Area of a square = Length^2

Thus, the length of the sign is 4x-5 inches

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