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3 years ago

An indirect proof has been successful if you reach a contradiction of a known fact.

Mathematics

2 answers:

harina [27]3 years ago

6 0

Answer:

True

Step-by-step explanation:

First we must understand the difference between direct and indirect proof

A direct proof assumes that the hypothesis of a conjecture is true, and then uses a logical well defined arguments to prove that the conclusion of the conjecture is true.

An indirect proof assumes that it is not true and hence relies on a contradiction to prove , and then running into a contradiction proving that the conjecture must be true.

Thus we assume a contradiction and proceed then we get a wrong conclusion stating that our assumption is false.

Thus a) True.

vlabodo [156]3 years ago

5 0

True if you really think about it start asking questions if you don't understand hope i help you good luck

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4 years ago

A 75-gallon tank is filled with brine (water nearly saturated with salt; used as a preservative) holding 11 pounds of salt in so

Debora [2.8K]

Let $A(t)$ = amount of salt (in pounds) in the tank at time $t$ (in minutes). Then $A(0) = 11$ .

Salt flows in at a rate

$\left(0.6\dfrac{\rm lb}{\rm gal}\right) \left(3\dfrac{\rm gal}{\rm min}\right) = \dfrac95 \dfrac{\rm lb}{\rm min}$

and flows out at a rate

$\left(\dfrac{A(t)\,\rm lb}{75\,\rm gal + \left(3\frac{\rm gal}{\rm min} - 3.25\frac{\rm gal}{\rm min}\right)t}\right) \left(3.25\dfrac{\rm gal}{\rm min}\right) = \dfrac{13A(t)}{300-t} \dfrac{\rm lb}{\rm min}$

where 4 quarts = 1 gallon so 13 quarts = 3.25 gallon.

Then the net rate of salt flow is given by the differential equation

$\dfrac{dA}{dt} = \dfrac95 - \dfrac{13A}{300-t}$

which I'll solve with the integrating factor method.

$\dfrac{dA}{dt} + \dfrac{13}{300-t} A = \dfrac95$

$-\dfrac1{(300-t)^{13}} \dfrac{dA}{dt} - \dfrac{13}{(300-t)^{14}} A = -\dfrac9{5(300-t)^{13}}$

$\dfrac d{dt} \left(-\dfrac1{(300-t)^{13}} A\right) = -\dfrac9{5(300-t)^{13}}$

Integrate both sides. By the fundamental theorem of calculus,

$\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac1{(300-t)^{13}} A\bigg|_{t=0} - \frac95 \int_0^t \frac{du}{(300-u)^{13}}$

$\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac{11}{300^{13}} - \frac95 \times \dfrac1{12} \left(\frac1{(300-t)^{12}} - \frac1{300^{12}}\right)$

$\displaystyle -\dfrac1{(300-t)^{13}} A = \dfrac{34}{300^{13}} - \frac3{20}\frac1{(300-t)^{12}}$

$\displaystyle A = \frac3{20} (300-t) - \dfrac{34}{300^{13}}(300-t)^{13}$

$\displaystyle A = 45 \left(1 - \frac t{300}\right) - 34 \left(1 - \frac t{300}\right)^{13}$

After 1 hour = 60 minutes, the tank will contain

$A(60) = 45 \left(1 - \dfrac {60}{300}\right) - 34 \left(1 - \dfrac {60}{300}\right)^{13} = 45\left(\dfrac45\right) - 34 \left(\dfrac45\right)^{13} \approx 34.131$

pounds of salt.

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2 years ago

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3 years ago

Read 2 more answers

What is the factored form of x^3 + 125?

klemol [59]

Answer:

(X+5)(x^2-5x+25)

Step-by-step explanation:

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4 years ago

What’s the equation for this table? Use y=mx+b. HELP

Verdich [7]

Answer:

y = 2x + 12

Step-by-step explanation:

You can check this by plugging any of the values from the table into the equation. You can find the answer by finding the slope and then using that to find the y-intercept.

y2-y1/x2-x1

16-14/2-1 = 2/1 = 2. So our slope equals 2.

If we set x = 0 to find our y-intercept, and we know that we have a slope of 2 we can subtract 2 from 14 to get the y-intercept.

14-2 = 12. --> (0, 12) which is true because we know that the slope is 2.

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3 years ago