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3 years ago

6.

Mathematics

1 answer:

photoshop1234 [79]3 years ago

7 0

Answer:

The answer is in the attached file below

Step-by-step explanation:

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1.26 is the price of one mile

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What is 12/3 as an improper fraction?

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Find the equation of a parabola that opens up and has the following x intercepts (-3,0) and (4,0)

Elenna [48]

The equation of a parabola that opens up and has the following x intercepts (-3,0) and (4,0) is $y=x^{2}-x-12$

<h3><u>Solution:</u></h3>

We have to find the equation of a parabola that opens up and has the following x intercepts (-3, 0) and (4, 0)

x-intercepts of the parabola are (−3, 0) and (4, 0)

So, we can form an equation:

Also x = 4

x – 4 = 0

x – 4 = 0 is another factor of quadratic equation.

The quadratic function is:

$\begin{array}{l}{y=(x+3)(x-4)} \\\\ {y=x^{2}-4 x+3 x-12} \\\\ {y=x^{2}-x-12}\end{array}$

Hence, the equation of the parabola is $y=x^{2}-x-12$

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3 years ago

<img src="https://tex.z-dn.net/?f=%5Cbegin%7Bequation%7D%5Ctext%20%7B%20Question%3A%20If%20%7D%20%5Cint_%7B%5Cfrac%7B-1%7D%7B%5C

Elodia [21]

We want to evaluate

$\displaystyle \int_{-\frac1{\sqrt2}}^{\frac1{\sqrt2}} \sqrt{\left(\frac{x-1}{x+1}\right)^2 + \left(\frac{x+1}{x-1}\right)^2 - 2} \, dx$

First we note that the integrand is even (replacing x with -x doesn't fundamentally alter the function being integrated), so this is equal to

$\displaystyle 2 \int_0^{\frac1{\sqrt2}} \sqrt{\left(\frac{x-1}{x+1}\right)^2 + \left(\frac{x+1}{x-1}\right)^2 - 2} \, dx$

The radicand reduces significantly to

$\displaystyle \left(\frac{x-1}{x+1}\right)^2 + \left(\frac{x+1}{x-1}\right)^2 - 2 = \frac{16x^2}{(1-x^2)^2}$

so that taking the square root, we simplify the integral to

$\displaystyle 8 \int_0^{\frac1{\sqrt2}} \frac x{1-x^2} \, dx$

which is trivially computed with a substitution of $y = 1 - x^2$ and $dy=-2x\,dx$ :

$\displaystyle -4 \int_1^{\frac12} \frac{dy}y = 4 \int_{\frac12}^1 \frac{dy}y = -4 \ln\left(\frac12\right) = \ln(2^4) = \boxed{\ln(16)}$

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2 years ago

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