A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after 1 s.

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Deufemiak7 · Accepted Answer

Step 1&amp;nbsp;The ripple travels outward at 60 cm/s, this is the rate of change of the radius with respect to time&amp;nbsp;dr&amp;nbsp;dt&amp;nbsp;=60cms&amp;nbsp;We can also derive an equation for the radius at time t from the problem description.r=60t&amp;nbsp;Step 2&amp;nbsp;The area of a circle is&amp;nbsp;A=πr2&amp;nbsp;, and we need to find the rate that this is increasing, which is&amp;nbsp;dA&amp;nbsp;dt&amp;nbsp;. So differentiate the A equation with respect to time t.&amp;nbsp;A=πr2&amp;nbsp;dA&amp;nbsp;dt&amp;nbsp;=π2rdr&amp;nbsp;dt&amp;nbsp;&amp;nbsp;We know the value of&amp;nbsp;dr&amp;nbsp;dt&amp;nbsp;&amp;nbsp;therefore, we can change that. Furthermore We can replace&amp;nbsp;r=60t&amp;nbsp;so that we have an equation where all we need to do to calculate the rate of change of the area is plug in the time t.dA&amp;nbsp;dt&amp;nbsp;=π2(60t)(60)=7200π&amp;nbsp;Step 3&amp;nbsp;So for t=1&amp;nbsp;dA&amp;nbsp;dt&amp;nbsp;=7200π(1)≈22619cm2s

madeleinejames20 · Accepted Answer

Step 1:To solve the problem, let&#039;s first define some variables:r - the radius of the circular ripple at time t in centimeters (cm).A - the area within the circle at time t in square centimeters (cm2).We are given that the ripple travels outward at a speed of 60 cm/s, which means that the rate of change of the radius with respect to time is constant and equal to 60 cm/s. Therefore, we can express the rate of change of the radius as:drdt=60 cm/s.We are asked to find the rate at which the area within the circle is increasing after 1 second, which means we need to find dAdt when t=1 second.The area of a circle is given by the formula: A=πr2.Differentiating both sides of this equation with respect to time t, we get:dAdt=2πrdrdt.Substituting the given value of drdt=60 cm/s, we have:dAdt=2πr·60 cm2/s.Step 2:Now, we need to find the value of r at t=1 second. Since the ripple is traveling at a constant speed, after 1 second, the radius of the ripple will be equal to the distance traveled, which is 60 cm/s multiplied by 1 second:r=60&amp;nbsp;cm/s·1&amp;nbsp;s=60 cm.Substituting this value into the equation for dAdt, we get:dAdt=2π·60cm·60&amp;nbsp;cm/s.Simplifying, we find:dAdt=7200π cm2/s.Therefore, the rate at which the area within the circle is increasing after 1 second is 7200π cm2/s.

Eliza Beth13 · Accepted Answer

Answer:7200πcm2/sExplanation:Given that the ripple travels outward at a speed of 60 cm/s, the radius of the circle will also increase at a rate of 60 cm/s.After 1 second, the radius will be r=60cm/s×1s=60cm.To find the rate of change of the area with respect to time, we differentiate the area formula with respect to time:dAdt=ddt(πr2)Using the chain rule, we can differentiate each term separately:dAdt=2πrdrdtSubstituting the values we have, we get:dAdt=2π×60cm×60cm/sSimplifying, we find:dAdt=7200πcm2/sTherefore, after 1 second, the area within the circle is increasing at a rate of 7200πcm2/s.

Mr Solver · Accepted Answer

The formula for the area of a circle is A=πr2. To find dAdt, we need to differentiate both sides of this equation with respect to time:dAdt=ddt(πr2)Using the chain rule, we can rewrite this as:dAdt=π·ddt(r2)To differentiate r2 with respect to time, we can apply the power rule:dAdt=π·2r·drdtGiven that the ripple is traveling outward at a speed of 60 cm/s, we can express drdt as 60 cm/s.Now, let&#039;s evaluate dAdt after 1 second. Since we&#039;re given the speed, we can find r after 1 second using the formula r(t)={speed}·t. Plugging in the values, we have r(1)=60{cm/s}·1{s}=60{cm}.Substituting r=60 cm and drdt=60 cm/s into the equation for dAdt, we get:dAdt=π·2·60{cm}·60{cm/s}Simplifying this expression gives us the rate at which the area within the circle is increasing after 1 second.Now, let&#039;s calculate the value:dAdt=π·2·60{cm}·60{cm/s}=7200π{cm}2/{s}Therefore, the rate at which the area within the circle is increasing after 1 second is 7200π{cm}2/{s}.

A stone is dropped into a lake, creating a circular

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