What is the missing constant term and a perfect square starts with x^2+10x

1 answer:

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The perfect square is a number that is the product of two identical numbers. So 9 is a perfect square because it's the product of 3 * 3. For variables, perfect squares can be expressed as $(x+a)^{2}$ or $(x-a)^{2}$ , where a is some number and x is the variable.
If we expand these terms, $(x+a)^{2}=x^{2}+2ax+a^{2}$ , and $(x-a)^{2}=x^{2}-2ax+a^{2}$ . Since the given term is $x^{2}+10x$ , we can see that it's part of the perfect square $x^{2}+2ax+a^{2}$ . We just need to find $a^{2}$ .
So, $x^{2}+2ax+a^{2}=x^{2}+10x+a^{2}$ . Notice that $10=2a$ , so $a=5$ . So, the missing constant term is $5^{2}=25$ . The perfect square is $(x+5)^{2}$ .

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Suppose a tank contains 400 gallons of salt water. If pure water flows into the tank at the rate of 7 gallons per minute and the

Strike441 [17]

Answer:

Step-by-step explanation:

This is a differential equation problem most easily solved with an exponential decay equation of the form

$y=Ce^{kt}$ . We know that the initial amount of salt in the tank is 28 pounds, so

C = 28. Now we just need to find k.

The concentration of salt changes as the pure water flows in and the salt water flows out. So the change in concentration, where y is the concentration of salt in the tank, is $\frac{dy}{dt}$ . Thus, the change in the concentration of salt is found in

$\frac{dy}{dt}=$ inflow of salt - outflow of salt

Pure water, what is flowing into the tank, has no salt in it at all; and since we don't know how much salt is leaving (our unknown, basically), the outflow at 3 gal/min is 3 times the amount of salt leaving out of the 400 gallons of salt water at time t:

$3(\frac{y}{400})$