Find two positive real numbers whose product is a maximum.
the sum of the first and three times the second is 30

15, 15

Step-by-step explanation:

Let the first and second number be x and y.

Three times the second number = 3y

As sum of the first number and three times the second number is 30,

x + 3y =30

we get, x = 30 - 3y

We need to find  two positive real numbers whose product is  maximum.

Let P =xy = y(30-3y)=

0n differentiating P with respect to y, we get

On putting , we get

0n differentiating P ' with respect to y, we get

At y = 5,

So, y = 5 is a point of maxima.

At y = 5,

So, the two positive numbers are 15, 15

step-by-step explanation:

the original amount doesn't ''re just looking for the time it takes for 1/2 of the substance to we have

(1/2) =)     take the ln   of both sides

ln (1/2)   = )   and we can write

ln(1/2) = ) lne     and lne   = 1     so we have

ln(1/2) = -.1092t     divide both sides by -.1092

t = ln(1/2)/-.1092 = about 6.347 days = about 6.3 days (rounded)

idk man good luck man