**Transcription for Educational Website:** --- **Problem Statement:** Suppose \( f(t) = 6(t - 4)^{-1/2} \). **(a) Find the derivative of \( f \).** \[ f'(t) = -3(t - 4)^{-3/2} \] *The derivative of the function \( f(t) \) is found using the power rule and chain rule of differentiation.* **(b) Find an equation for the tangent line to the graph of \( y = f(t) \) at the point \( (t, y) = (29, 6/5) \).** *Tangent line:* \[ y = -\frac{3}{125} x + \frac{237}{125} \] *This equation represents the tangent line at the specified point. The slope is calculated using the derivative \( f'(t) \), and the equation is found using the point-slope form.* --- **Additional Information:** In the provided answer section, the values for the derivative and tangent line are confirmed through a second attempt, indicated as "Attempt 2 of 2". *Note: There was an alert symbol (⚠) next to the equation of the tangent line, possibly indicating an error or a point of caution in a digital platform, which should be checked for accuracy.*

There is a partial insurance policy that pays $80,000 upon the bad droughtstate (and $0 in the normal, no drought state). Let Z be the most farmer Brown would be willing to pay forsuch a policy. Carefully set up the equation that defines Z, along with the appropriate captions or sentencethat explains your equation.

Type an equation iny=mxform.

During Ricardo noticed an ink stain with a diameter of about 3 cm on the front of his teacher’s shirt. He then noticed that there was a pen in the pocket and that the ink blot was continuing to spread across the front of the teacher’s shirt. He estimated that the diameter of the ink blot seemed to be growing by about 0.5 cm every 2 minutes. a) Write an equation for the RADIUS, r, of the ink stain, as a function of time, t. Assume that t = 0 represents the time that Ricardo first noticed the ink stain. b) Write an equation showing the AREA, A, of the ink stain as a function of the time t. (Hint: It might help to first write A as a function of r. ) c) Draw a graph of the function A(t) for 10 minutes. d) At what time is the area of the ink stain about 25 square centimeter? Show how you answer this question. (i.e. ”I found it using desmos” is not sufficient)