After a robot vacuum is fully charged, it can run for 15 minutes. It can vacuum 1/2 square foot every second. Can a fully charged robot clean a room that is 400 square feet without recharging?

The robot can clean the room without recharging.

Step-by-step explanation:

Given:

After a robot vacuum is fully charged, it can run for minutes.

The robot cleans square foot every second.

To find: If the charged robot can clean a room that is square feet without recharging.

Solution:

Area cleaned by the robot in second square foot.

We know that, minute seconds.

So, minutes seconds.

Now, area cleaned by the robot in seconds square feet.

The area to be cleaned is square feet.

Hence, the robot can clean the room without recharging.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!