wall A ladder 10 ft long rests against a vertical wall. If the bottom of the lad- der slides away from the wall at a rate of I ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? SOLUTION We first draw a diagram and label it as in Figure 1. Let & feet be the dis- tance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground. Note that & and y are both functions of / (time, measured in seconds). We are given that dx/di - fit/s and we are asked to find dy/ di when x -Oft (see ground Figure 2). In this problem, the relationship between x and y is given by the Pythagorean FIGURE 1 Theorem: Differentiating each side with respect to / using the Chain Rule, we have 2x d.x dy - + 2y it - 0 and solving this equation for the desired rate, we obtain dy dx When x = 6. the Pythagorean Theorem gives y = 8 and so, substituting these values FIGURE 2 and dx/di - 1, we have dy () - -3 A/S The fact that dy/ dr is negative means that the distance from the top of the ladder to the ground is decreasing at a rate of # ft/s. In other words, the top of the ladder is sliding down the wall at a rate of i fi/s. What is the value in the red box? What is the value in the blue box? What is the value in the green box? What is the value in the pink box? What is the value in the orange box? I confirm that I know how to differentiate these types of functions (yes or no)Recall the definition of Hyperbolic Functions: Definition of the Hyperbolic Functions sinh x = csch X = 2 sinh x e'te cosh x = sech x = 2 cosh x sinh x tanh X = cosh x coth * = cosh x sinh x Hyperbolic Identities sinh(-x) - - sinh x cosh(-x) - cosh x cosh'x - sinh x - 1 1 - tanhex = sech x sinh(x + y) = sinh x cosh y + cosh x sinh y cosh(x + y) = cosh x cosh y + sinh x sinh y Now answer the following question: coshar - sinher = 24 -2+el 4 4 = What is the value in the red box? What is the value in the blue box? What is the value in the green box? What is the value in the pink box? What is the value in the orange box? I confirm that I know how to differentiate these types of functions (yes or no)Find the critical numbers of ((x) The first value is The second value isFind the absolute maximum and minimum values of the function Absolute maximum is Absolute minimum is