Question

A cart rolls down a hill, traveling $5$ inches the first second and accelerating so that during each successive $1$ -second time interval, it travels $7$ inches more than during the previous $1$ -second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel?

$215$

$360$

$2992$

$3195$

$3242$

Video solution:

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Written solution:

The distance travelled every second forms an arithmetic sequence: $5, 5 + 7, 5 + 2 \cdot 7, \ldots$

The sum of an arithmetic sequence is \text{# of terms} \cdot \dfrac{\text{first + last}}{2}. We know the number of terms is $30$ and the first term is $5.$ The last term is $5 + 29 \cdot 7 = 208.$

Plugging these values into the expression yields $30 \cdot \dfrac{5 + 208}{2} = 15 \cdot 213 = 3195.$

Thus, D is the correct answer.

2021 AMC 10A Spring Problem 4

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Problem 4 in Other Years