Question

a drop of water hits the surface of a lake and forms concentric circular ripples. the radius, in inches, of the circle enclosed by the outer ripple increases as a function of time ( t ), in minutes, and is modeled by the function ( r ) given by ( r(t)=25sqrt{2t + 1} ). which of the following gives the area of the circle enclosed by the outer ripple, in square inches, as a function of time ( t ), in minutes? (note: the area of a circle with radius ( x ) is given by ( a=pi x^{2} ).)

Explanation:

Step1: Recall the formula for the area of a circle

The area formula of a circle is \(A=\pi x^{2}\), where \(x\) is the radius. Here, \(x = r(t)=25\sqrt{2t + 1}\).

Step2: Substitute the radius function into the area formula

Substitute \(r(t)\) into \(A\):

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Step3: Simplify the expression

Use the property \((ab)^{2}=a^{2}b^{2}\) and \((\sqrt{m})^{2}=m\) (\(m\geq0\)).

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Answer:

\(A(t)=1250\pi t + 625\pi\)