A BILLBOARD PAINTER HAS BEEN ASSIGNED THE TASK OF CHANGING THE ADVERTISMENT ON A 20-FT BILLBOARD,THE BOTTOM OF WHICH IS 15FT OFF THE GROUND.AFTER EXAMINING THE SITE, SHE IDENTIFIES TWO AREAS ON THE GROUNG UNDER THE SIGN THAT ARE STURDY ENOUGH TO SUPPORT HER LADDER. ONE ARE IS 10-FT FROM THE BASE OF THE BILLBOARD AND THE OTHER IS 15-FT FROM THE BASE..
To determine which location would require the smallest ladder, we need to compare the distances from the base of the billboard to each area on the ground where the ladder can be placed.
Let's calculate the length of the ladder in each case:
1. For the area that is 10ft from the base of the billboard:
We have a right-angled triangle with the ground as the base, the height as the distance from the ground to the bottom of the billboard, and the ladder as the hypotenuse.
Using the Pythagorean theorem, we can find the length of the ladder:
ladder^2 = ground^2 + height^2
ladder^2 = 10^2 + 15^2
ladder^2 = 100 + 225
ladder^2 = 325
ladder ≈ √325
ladder ≈ 18.03 ft
2. For the area that is 15ft from the base of the billboard:
Again, we have a right-angled triangle with the ground as the base, the height as the distance from the ground to the bottom of the billboard, and the ladder as the hypotenuse.
Using the Pythagorean theorem:
ladder^2 = ground^2 + height^2
ladder^2 = 15^2 + 15^2
ladder^2 = 225 + 225
ladder^2 = 450
ladder ≈ √450
ladder ≈ 21.21 ft
Therefore, based on these calculations, the billboard painter would need a ladder of approximately 18.03 ft if she places it 10 ft from the base of the billboard, whereas she would need a ladder of approximately 21.21 ft if she places it 15 ft from the base. So, using the area that is 10 ft from the base of the billboard would require a smaller ladder.