Question

the velocity of a car is ( f(t) = 7t ) meters/second. use a graph of ( f(t) ) to find the exact distance traveled by the car, in meters, from ( t = 0 ) to ( t = 10 ) seconds. distance = (include units)

Explanation:

Step1: Analyze the velocity - time graph

The velocity function is \(f(t)=7t\), which is a linear function. The graph of \(y = f(t)\) is a straight - line passing through the origin with a slope of \(7\). The distance traveled by the car from \(t = a\) to \(t = b\) is given by the area under the velocity - time graph from \(t=a\) to \(t = b\). For \(a = 0\) and \(b = 10\), the area under the graph of \(y=7t\) from \(t = 0\) to \(t = 10\) forms a right - triangle.

Step2: Use the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). In the context of the velocity - time graph, the base of the triangle is the time interval \(\Delta t=b - a\), where \(a = 0\) and \(b = 10\), so \(base=10-0 = 10\) seconds. The height of the triangle is the value of the velocity function at \(t = 10\), so \(f(10)=7\times10=70\) meters/second.

Step3: Calculate the area

Substitute the base and height into the area formula: \(A=\frac{1}{2}\times10\times70\).

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Answer:

\(350\) meters