Suppose f(t) is the weight (in grams) of a solid sitting in a beaker of water and the solid is dissolving at a rate (in grams/minute) of f'(t) =-4 f(t)(3 + f(t) ).

If there is 4 grams of solid at time t=2 minutes, use a linear approximation to estimate the solid's weight 1 second later.

To estimate the solid's weight 1 second later, we can use linear approximation, also known as tangent line approximation. Here's how you can do it step by step:

Step 1: Find the value of f(t) at t = 2 minutes
Given that there are 4 grams of solid at t = 2 minutes, we can say that f(2) = 4.

Step 2: Find the derivative of f(t)
The given rate of dissolution is f'(t) = -4f(t)(3 + f(t)). We need the derivative in order to find the tangent line.

Taking the derivative of f(t) with respect to t:
f'(t) = -4f(t)(3 + f(t))

Step 3: Find the slope of the tangent line at t = 2 minutes
Evaluate the derivative we found in step 2 at t = 2 minutes:
f'(2) = -4f(2)(3 + f(2))

Substitute f(2) = 4 into the equation:
f'(2) = -4(4)(3 + 4)
f'(2) = -4(4)(7)
f'(2) = -112

So, the slope of the tangent line at t = 2 minutes is -112 grams/minute.

Step 4: Use the point-slope formula to find the equation of the tangent line
The equation of a line in point-slope form is given by:
y - y1 = m(x - x1)

Substitute the values we have:
m = -112 (slope)
x1 = 2 minutes (given time)
y1 = 4 grams (given weight)

The equation of the tangent line is:
y - 4 = -112(t - 2)

Step 5: Convert the time to seconds
There are 60 seconds in 1 minute. So, 2 minutes is equal to 2 * 60 = 120 seconds.

Step 6: Estimate the solid's weight after 1 second
To find the solid's weight 1 second later, substitute t = 120 + 1 = 121 seconds into the equation of the tangent line:
y - 4 = -112(121 - 2)
y - 4 = -112(119)
y - 4 = -13328
y = -13328 + 4
y = -13324 grams

Therefore, the estimated weight of the solid 1 second later is approximately -13324 grams. Note that the negative sign indicates a decrease in weight due to dissolution.