Final answer:
To find the length of the ladder Antoine should buy, we can use trigonometry and the sine function. By rearranging the equation, we can solve for the length of the ladder, which is found to be 33.15 feet.
Step-by-step explanation:
To calculate the length of the ladder Antoine should buy, we can use trigonometry. Given that the base of the ladder makes a 51.8 ° angle with the ground and Antoine wants the ladder to reach at least 27 feet high, we can use the sine function to find the length of the ladder. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the height of the ladder and the hypotenuse is the length of the ladder.
So we have:
sin(51.8 °) = height of ladder / length of ladder
By rearranging the equation, we can solve for the length of the ladder:
length of ladder = height of ladder / sin(51.8 °)
Plugging in the values, we get:
length of ladder = 27 feet / sin(51.8 °) = 33.15 feet
Therefore, Antoine should buy a ladder that is at least 33.15 feet long.