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3 years ago

What are the properties of the incenter of a triangle

Mathematics

1 answer:

Anit [1.1K]3 years ago

5 0

<span>Located at intersection of the angle bisectors. See Triangle incenter definition</span>

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Help calculus module 8 DBQ<br><br> please show work

igor_vitrenko [27]

1. The four subintervals are [0, 2], [2, 3], [3, 7], and [7, 8]. We construct trapezoids with "heights" equal to the lengths of each subinterval - 2, 1, 4, and 1, respectively - and the average of the corresponding "bases" equal to the average of the values of $R(t)$ at the endpoints of each subinterval. The sum is then

$\dfrac{R(0)+R(2)}2(2-0)+\dfrac{R(2)+R(3)}2(3-2)+\dfrac{R(3)+R(7)}2(7-3)+\dfrac{R(7)+R(8)}2(7-8)=\boxed{24.83}$

which is measured in units of gallons, hence representing the amount of water that flows into the tank.

2. Since $R$ is differentiable, the mean value theorem holds on any subinterval of its domain. Then for any interval $[a,b]$ , it guarantees the existence of some $c\in(a,b)$ such that

$\dfrac{R(b)-R(a)}{b-a)=R'(c)$

Computing the difference quotient over each subinterval above gives values of 0.275, 0.3, 0.3, and 0.26. But just because these values are non-zero doesn't guarantee that there is definitely no such $c$ for which $R'(c)=0$ . I would chalk this up to not having enough information.

3. $R(t)$ gives the rate of water flow, and $R(t)\approx W(t)$ , so that the average rate of water flow over [0, 8] is the average value of $W(t)$ , given by the integral

$R_{\rm avg}=\displaystyle\frac1{8-0}\int_0^8\ln(t^2+7)\,\mathrm dt$

If doing this by hand, you can integrate by parts, setting

$u=\ln(t^2+7)\implies\mathrm du=\dfrac{2t}{t^2+7}\,\mathrm dt$

$\mathrm dv=\mathrm dt\implies v=t$

$R_{\rm avg}=\displaystyle\frac18\left(t\ln(t^2+7)\bigg|_{t=0}^{t=8}-\int_0^8\frac{2t^2}{t^2+7}\,\mathrm dt\right)$

For the remaining integral, consider the trigonometric substitution $t=\sqrt 7\tan s$ , so that $\mathrm dt=\sqrt 7\sec^2s\,\mathrm ds$ . Then

$R_{\rm avg}=\displaystyle\ln71-\frac{\sqrt7}4\int_0^{\tan^{-1}(8/\sqrt7)}\frac{7\tan^2s}{7\tan^2s+7}\sec^2s\,\mathrm ds$

$R_{\rm avg}=\displaystyle\ln71-\frac{\sqrt7}4\int_0^{\tan^{-1}(8/\sqrt7)}\tan^2s\,\mathrm ds$

$R_{\rm avg}=\displaystyle\ln71-\frac{\sqrt7}4\int_0^{\tan^{-1}(8/\sqrt7)}(\sec^2s-1)\,\mathrm ds$

$R_{\rm avg}=\displaystyle\ln71-\frac{\sqrt7}4\left(\tan s-s\right)\bigg|_{s=0}^{s=\tan^{-1}(8/\sqrt7)}$

$R_{\rm avg}=\displaystyle\ln71-\frac{\sqrt7}4\left(\tan\left(\tan^{-1}\frac8{\sqrt7}\right)-\tan^{-1}\frac8{\sqrt7}\right)$

$\boxed{R_{\rm avg}=\displaystyle\ln71-2+\frac{\sqrt7}4\tan^{-1}\frac8{\sqrt7}}$

or approximately 3.0904, measured in gallons per hour (because this is the average value of $R$ ).

4. By the fundamental theorem of calculus,

$g'(x)=f(x)$

and $g(x)$ is increasing whenever $g'(x)=f(x)>0$ . This happens over the interval (-2, 3), since $f(x)=3$ on [-2, 0), and $-x+3>0$ on [0, 3).

5. First, by additivity of the definite integral,

$\displaystyle\int_{-2}^xf(t)\,\mathrm dt=\int_{-2}^0f(t)\,\mathrm dt+\int_0^xf(t)\,\mathrm dt$

Over the interval [-2, 0), we have $f(x)=3$ , and over the interval [0, 6], $f(x)=-x+3$ . So the integral above is

$\displaystyle\int_{-2}^03\,\mathrm dt+\int_0^x(-t+3)\,\mathrm dt=3t\bigg|_{t=-2}^{t=0}+\left(-\dfrac{t^2}2+3t\right)\bigg|_{t=0}^{t=x}=\boxed{6+3x-\dfrac{x^2}2}$

6 0

3 years ago

An old bone contains 80% of its original carbon-14. Use the half-life model to find the age of the bone. Find an equation equiva

Ira Lisetskai [31]

Answer:

An old bone contains 80% of its original carbon-14 in 1844.6479 years

Step-by-step explanation:

We know that

half life time of C-14 is 5730 years

so, h=5730

now, we can use formula

$P(t)=P_0(\frac{1}{2})^{\frac{t}{h} }$

we can plug back h

and we get

$P(t)=P_0(\frac{1}{2})^{\frac{t}{5730} }$

An old bone contains 80% of its original carbon-14

so,

P(t)=0.80Po

we can plug it and then we solve for t

$0.80P_0=P_0(\frac{1}{2})^{\frac{t}{5730} }$

$0.80=(\frac{1}{2})^{\frac{t}{5730} }$

$\ln \left(0.8\right)=\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{5730}}\right)$

$t=-\frac{5730\ln \left(0.8\right)}{\ln \left(2\right)}$

$t=1844.6479$

So,

An old bone contains 80% of its original carbon-14 in 1844.6479 years

4 0

3 years ago

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Unite price batteries size 4pk and total price 2.97 how come unite price

omeli [17]

To find the unit price, divide the unit by the the total amount:

2.97/4
0.7425

Round this to the nearest cent:

$0.74

Hope this helps!

5 0

3 years ago

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Alguien me puede ayudar porfas Osino repito

Svetach [21]

Answer:

Translate?

Step-by-step explanation:

3 0

3 years ago

Which term can be used in the blank of 36x3−22x2−__ so the greatest common factor of the resulting polynomial is 2x? Select two

maria [59]

Answer:

4xy and 12x

Step-by-step explanation:

In this question, we are to select which of the options to put into the blank space that will make the greatest common factor of the polynomial 2x. We are to select two options

Now, looking at the blank space, we should understand that , any answer we shall be choosing shall be an answer that contains x alongside the number.

This makes 24 and 44y out of the way.

Now let’s look at 4xy and 12x. Let’s insert them into the blank space;

36x^3-22x^2 - 4xy which gives : 2x(18x^2-11x-2y)

and 36x^3-22x^2-12x = 2x(18x^2-11x-6)

The reason why we will not use 44y and 24 is because 2x is not a factor of either.

3 0

3 years ago

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