Consider the initial value problem dy dt = −t + (t 2 + 4y) 1/2 2 , y(2) = −1. (a) Verify that both y 1 (t) = 1 − t and y 2 (t) = −t 2 /4 are both solutions to the initial value problem. (b) Explain why the existence of two solutions doesn’t violate the uniqueness part of the existence and uniqueness theorems discussed in class. (c) Show that for any c ∈ R, the function y(t) = ct + c 2 is a solution to the differential equation for t ≥ −2c; that if c = −1, then the solution y(t) satisfies the initial condition, and that in this case y(t) = y 1 (t); and that there is no choice of c that makes y(t) equal to y 2 (t)

When keisha installed a fence along the 200 foot perimeter of her rectangular back yard, she left an opening for a gate. in the diagram below, she used x to represent the length in feet of the gate? what is the value? a. 10 b. 20 c. 25 d. 30

What is the x-intercept and y-intercept of the equation 2x+1. 2o

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?