Find the greatest possible value of the product xy, given that x and y are both positive and x + 2y = 30

The maximum value of the product x*y is 112.5

Step-by-step explanation:

We have that x + 2y = 30

so we can isolate one of the variables and get:

x = 30 - 2y

now, the product of x*y can be written as:

x*y = (30 - 2y)*y = 30y - 2y^2

now, we want to find the maximum of 30y - 2y^2

because the leading coeficient is negative (-2), we know that the hands of the quadratic function will go downwards, so the zero of the derivate of this quadratic equation will give us the value where the equation has a maximum:

f(y) = -2*y^2 + 30*y

f'(y) = -4*y + 30 = 0

y =  -30/4 = 7.5

So the maximum value is:

f(7.5) = -2*7.5^2 + 30*7.5 = 112.5

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."

overpriced

step-by-step explanation: