h a = 3 + 28.6 A pediatrician uses the model above to estimate the height h of a boy, in inches, in terms of the boy’s age a, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, of a boy’s height each year? A) 3 B) 5.7 C) 9.5 D) 14.3
Choice A is correct. In the equation h = 3a + 28.6, if a, the age of the boy, increases by 1, then h becomes h = 3(a + 1) + 28.6 = 3a + 3 + 28.6 = (3a + 28.6) + 3. Therefore, the model estimates that the boy’s height increases by 3 inches each year. Alternatively: The height, h, is a linear function of the age, a, of the boy. The coefficient 3 can be interpreted as the rate of change of the function; in this 28 case, the rate of change can be described as a change of 3 inches in height for every additional year in age. Choices B, C, and D are incorrect and are likely to result from common errors in calculating the value of h or in calculating the difference between the values of h for different values of a.
True/False? Steven only consumes two goods: X and Y. If X is a Giffen good for Steven, then Y must be a normal good for Steven.
True. A Giffen good is an inferior good. Since Steven only consumes two goods, they cannot both be inferior goods. Therefore, Y is a normal good. (Need to point out explicitly that Giffen goods are a special kind of inferior goods, or the negative income effect of a Giffen good dominates the substitution effect. People who only say that the consumption of a Giffen good decreases with income without explanation lose partial points.)
Find the vertex, the focus, the axis of symmetry and the directrix of the parabola defined by the equation x 2 - 8x - y + 2 = 0
We first complete the square using the terms in x and x 2 and write the given equation in the form (x - h) 2 = 4a (y - k) where (h , k) is the vertex and the focus is at (h , k + a), the axis of symmetry is given by x = h and the directrix is given by y = k - a x 2 - 8x - y + 2 = 0 ((x - 4) 2 - 16) - y + 2 = 0 (x - 4) 2= (y + 14) vertex at (4 , -14) 1 = 4a , hence a = 1/4 focus at (4 , -14 + 1/4) = (4 , -13.75) axis of symmetry given by x = 4 Directrix is a horizontal line given by: y = k - a = - 14 - 1/4 = -14.25