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3 years ago

Prove that the product of a nonzero rational number together with an irrational number is an irrational number.

1 answer:

7 0

Each time they assume the sum is rational; however, upon rearranging the terms of their equation, they get a contradiction (that an irrational number is equal to a rational number). Since the assumption that the sum of a rational and irrational number is rationalleads to a contradiction, the sum must be irrational.

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Need help #3. The answer is shown, but I don’t know how to get to the answer. Please teach and show steps.

Sunny_sXe [5.5K]

Answer:

Step-by-step explanation:

We are given a right triangle with a base of x feet and a height of h feet, where x is constant and h changes with respect to time t.

The angle in radians is defined by:

$\displaystyle \tan(\theta)=\frac{h}{x}$

And we want to find the relationship that describes dθ/dt and dh/dt.

So, we will differentiate both sides with respect to t where x is a constant:

$\displaystyle \frac{d}{dt}[\tan(\theta)]=\frac{d}{dt}\Big[\frac{h}{x}\Big]$

Differentiate. Apply the chain rule on the left. Again, remember that x is just a constant, so we can move it outside the derivative operator. Therefore:

$\displaystyle \sec^2(\theta)\frac{d\theta}{dt}=\frac{1}{x}\frac{dh}{dt}$

Since we know that tan(θ)=h/x, h is the opposite side of our triangle and x is the adjacent. Therefore, by the Pythagorean Theorem, our hypotenuse will be:

$\text{Hypotenuse}=\sqrt{h^2+x^2}$

Since secant is the ratio of the hypotenuse to adjacent:

$\displaystyle \sec(\theta)=\frac{\sqrt{h^2+x^2}}{x}$

So:

$\displaystyle \sec^2(\theta)=\frac{x^2+h^2}{x^2}$

By substitution, we have:

$\displaystyle \Big(\frac{x^2+h^2}{x^2}\Big)\frac{d\theta}{dt}=\frac{1}{x}\frac{dh}{dt}$

By multiplying both sides by the reciprocal of the term on the left:

$\displaystyle \frac{d\theta}{dt}=\frac{1}{x}\Big(\frac{x^2}{x^2+h^2}\Big)\frac{dh}{dt}$

Therefore:

$\displaystyle \frac{d\theta}{dt}=\frac{x}{x^2+h^2}\frac{dh}{dt}$

Our answer is A.

3 0

3 years ago

In engineering, the modulus of elasticity is a way to measure how much an object deforms along an axis when opposing forces are

Allushta [10]

Step-by-step explanation:

Consider an engineering material of initial length Lo, Area (A), Modulus of elasticity (E) and applied a force P due to which change in the length of the material is δ2 from it’s original length (Lo)

Initial length of the material is Lo. Hence, at time t = 0 when no force applied on the material the length of the material will not change (i.e., at time t=0, δ1 = 0)

Modulus of elasticity of the material:

$E=\frac{P \cdot L_{o}}{A\left[\delta_{2}-\delta_{1}\right]}$

Area of the material:

$E=\frac{P \cdot L_{o}}{A\left[\delta_{2}-\delta_{1}\right]}$

$A=\frac{P \cdot L_{o}}{E\left[\delta_{2}-\delta_{1}\right]}$

Length of the material:

$E=\frac{P \cdot L_{0}}{A\left[\delta_{2}-\delta_{1}\right]}$

$L_{0}=\frac{E \cdot A\left[\delta_{2}-\delta_{1}\right]}{P}$

5 0

3 years ago

In an

Usimov [2.4K]

Answer:

8 games in a row

Step-by-step explanation:

If they win 8 games in a row, there winning percentage will be 20/33, which is equal to about 60%.

3 0

3 years ago

Please you can help me. Thanks.

nikdorinn [45]

Answer:

Using Math Papa Algebra Calculator I was able to simplify to:

-81x^5+1198x^3-896x=0

In further detail this equation can be factored and then solved for the multiple x-values given, which are: x=0 or x=−3.7416573867739413 or x=3.7416573867739413 or x= -8/9 or x=8/9

Step-by-step explanation:

I used a punnet square to solve for the simplified equation, from there I used Math Papa Algebra Calculator.

I would answer with the simplified answer, sorry I'd need more specifics to answer this question fully.

Once again the simplified is -81x^5+1198x^3-896x=0

3 0

3 years ago

Write the equation of a line through (1,2), parallel to y=4x+1

AveGali [126]

Answer:

Step-by-step explanation:

y - 2 = 4(x - 1)

y - 2 = 4x - 4

y = 4x - 2

4 0

4 years ago