Write the number 270 as a sum of three numbers so that the sum of the products taken two at a time is a maximum. (Enter the three numbers as a comma-separated list.)

x=y=z=90.

Step-by-step explanation:

let the 3 numbers be x,y,z.

let s be sum

s=x+y+z

we need to maximize xy+yz+zx

or , xy + y(270-(x+y))+x(270-(x+y)) = P(x)

Use Lagrange's Multipliers

⇒ΔP= λΔS

⇒

⇒y=x

⇒\frac{x+z}{x+y} =1

⇒z=y

⇒x=y=z

⇒3x = 270

⇒x= 90

hence, x=y=z=90.

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.