A substance with a half life is decaying exponentially. If there are initially 12grams of the substance and after 70 minutes there are 7 grams, after how many minutes will there be 2 grams remaining?

Step-by-step explanation:

Use the half life formula

where N is the amount after decay, No is the initial amount, e is Euler's number, k is the decay constant, and t is the time in minutes. For us, the first equation looks like this:

and we will solve that for k and then use that value of k in the second equation to find time.

Begin by dividing 7 by 12 to get

Now take the natural log of both sides since the natural log and that e are inverses. The e then disappears:

Plug the left side into your calculator and then set it equal to the right side, giving us:

Divide both sides by 70 to get the k value of

k = -.00769995

Now the second equation looks like this:

where t is our only unknown now that we know k.

Begin by dividing both sides by 12 to get

and take the natural log of both sides again to eliminate the e:

Take care of the left side on your calculator to get

-1.791759469 = -.00769995t and divide both sides by -.00769995 to get

t = 232.697 minutes

answer: 7

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240