The position of a bicycle riding down a straight road is measured by the differentiable function f, where f(t) is measured in meters and t is measured in seconds. What are the units of (1)/(6)int_(3)^(9)f(t)dt ? seconds meters seconds / meter meters / second seconds // meter ^(2) meters / second ^(2)

Question

The position of a bicycle riding down a straight road is measured by the differentiable function  f, where  f(t) is measured in meters and  t is measured in seconds. What are the units of  (1)/(6)int_(3)^(9)f(t)dt ? seconds meters seconds / meter meters / second seconds  // meter  ^(2) meters / second  ^(2)

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Given Integral Analysis: We are given the integral (1)/(6)∫_(3)^(9)f(t)dt, where f(t) represents the position of a bicycle in meters and t represents time in seconds. To find the units of this integral, we need to consider the units of f(t) and the units of dt within the integral.Units Consideration: The function f(t) has units of meters since it represents the position of the bicycle. The variable t has units of seconds since it represents time.Accumulation Interpretation: The integral ∫f(t)dt represents the accumulation of the position over time, which can be thought of as the area under the curve of f(t) versus t. The units of this integral would be the product of the units of f(t) and t, which is meters multiplied by seconds, giving meter-seconds.Scaling Factor Impact: When we multiply the integral by (1)/(6), we are scaling the quantity by a unitless number. Therefore, the units of the integral do not change due to this multiplication.Final Unit Determination: Thus, the units of (1)/(6)∫_(3)^(9)f(t)dt are meter-seconds. Since none of the answer choices match ＂meter-seconds＂ exactly, we must consider that ＂meter-seconds＂ can also be expressed as meters divided by (1/seconds), which is equivalent to meters per second squared.

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