A large asteroid crashed into a moon of another planet causing several boulders from the moon to be propelled into space toward the planet. Astronomers were able to measure the speed of one of the projectiles.

Question

The distance (in feet) that the projectile traveled each second, starting with the first second, was given by the arithmetic progression 17 comma 51 comma 85 comma 119 comma ellipsis Find the distance that the projectile traveled in the ninth second.
(1 point)
Responses

306 ft.
306 ft.

323 ft.
323 ft.

255 ft.
255 ft.

289 ft.
289 ft.

Answer 1

The distance traveled by the projectile in each second forms an arithmetic progression (AP). The series starts with the first term \( a_1 = 17 \) and has a common difference \( d \) that can be calculated from the first two terms.

To find the common difference \( d \): \[ d = a_2 - a_1 = 51 - 17 = 34 \]

Now, we can express the \( n \)-th term of an arithmetic progression using the formula: \[ a_n = a_1 + (n-1) \cdot d \]

We want to find the distance traveled in the ninth second, which corresponds to \( n = 9 \): \[ a_9 = a_1 + (9-1) \cdot d \] \[ a_9 = 17 + 8 \cdot 34 \] \[ a_9 = 17 + 272 \] \[ a_9 = 289 \]

Thus, the distance traveled by the projectile in the ninth second is \( 289 \) feet.

The correct response is: 289 ft.