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Mathematics

1 answer:

Masteriza [31]3 years ago

8 0

Answer:

Shifted by 6 units left on the x-axis.

Step-by-step explanation:

Given function is,

f(x) = 5ˣ + 1

After transformation, this function becomes g(x) = $5^{(x+6)}+1$

Here, g(x) = f(x + 6)

By the rule of transformation,

Function 'f' has been shifted by 6 units left on the x-axis to form a new function 'g'.

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Robert and Chris can skate 4 miles in 18 minutes. How far can they ride in 45 minutes?

adell [148]

Answer: Robert and Chris can skate 10 miles in 45 minutes.

Step-by-step explanation:

Let's start by finding the unit rate.

$\frac{4}{18} =\frac{2}{9} \\\frac{18}{18} =1$

Robert and Chris can skate $\frac{2}{9}$ of a mile in 1 minute. Now we can multiply both by 45 to see how far the can go in 45 minutes.

$(\frac{2}{9}) (\frac{45}{1} )\\\frac{(2)(45)}{(9)(1)} \\\frac{90}{9} \\10$

45x1=45

The can skate 10 miles in 45 minutes.

6 0

3 years ago

How to find the average squared distance between the points of the unit disk and the point (1,1)

ddd [48]

The unit disk can be parameterized by the function

$\mathbf p(r,\theta)=(r\cos\theta,r\sin\theta)$

where $0\le r\le 1$ and $0\le\theta\le2\pi$ . The squared distance between any point in this region $(x,y)=(r\cos\theta,r\sin\theta)$ and the point (1, 1) is

$(x-1)^2+(y-1)^2=(r\cos\theta-1)^2+(r\sin\theta-1)^2$
$=(r^2\cos^2\theta-2r\cos\theta+1)+(r^2\sin^2\theta-2r\sin\theta+1)$
$=r^2(\cos^2\theta+\sin^2\theta)-2r(\cos\theta-\sin\theta)+2$
$=r^2-2r(\cos\theta-\sin\theta)+2$

The average squared distance is then going to be the ratio of [the sum of all squared distances between every point in the disk and the point (1, 1)] to [the area of the disk], i.e.

$\dfrac{\displaystyle\iint_{x^2+y^2$
$=\dfrac{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}[r^2-2r(\cos\theta-\sin\theta)+2]r\,\mathrm dr\,\mathrm d\theta}{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}r\,\mathrm dr\,\mathrm d\theta}$
$=\dfrac{\frac{5\pi}2}\pi=\dfrac52$