Consider the following conditions Two positive numbers have

Consider the following conditions: Two positive numbers have the property that one less than the first number, and three more than twice the second, totals to 50. (i) If we call the first number x, and the second z, write an equation that relates these numbers. (ii) Use (i) to write an equation that gives the second in terms of the first. (iii) Use (ii) to write an expression for y=f (x), the product of these two numbers, in terms of x. (iv) Now write the product formula in vertex form, by using the completing the square technique. (v) Use (iv) to graph the function in (iii). Label the coordinates of the maximum product, as well as both x-intercepts. (vi) Find both numbers, as well as the largest product. (vii) Write a complete sentence explanation of the answers to (vi). (viii) Solve 2 cos^2x + 5sinx = 4 for ALL possible x. (ix) In a complete English sentence, explain why this apparently unrelated question appears here in problem #2.

Solution

2. i. x-1+2z+3=50

x+2z= 48

ii. x+2z=48

z= (48-x)/2

iii. y=xz= x(48-x)/2

iv. y= -1/2(x^2-48x +576-576) = (-1/2)(x-24)² +288

vi. From the graph,we can say that x intercepts are(0,0) and (48,0). And the largest product is at the vertex which is 288

v.