Mathematical Models: Building Functions Q1,2,3,5

1. Let P= (x,y) be a point on the graph of y = x2 - 2. (a) Express the distance d from P to the origin as a function of x. d(x) = (b) What is d if x = 0? d(0) = (Round to two decimal places.) (c) What is d if x = 1? d(1) = (Round to two decimal places.) Use a graphing utility to graph d = d(x) in order to answer the following question. (d) For what positive value of x is d smallest? X= (Round to two decimal places.)2. A rectangle has one corner in quadrant I on the graph of y = 9 - x, another at the origin, a third 12 Ty on the positive y-axis, and the fourth on the positive x-axis (see the figure). (a) Express the area A of the rectangle as a function of x. (b) What is the domain of A? (c) Graph A = A(x). For what value of x is A largest? 3 (d) What is the largest area? (0,0 2 -3 (a) The area A of the rectangle as a function of x is A(x) = (Simplify your answer.) (b) The domain is (Type your answer in interval notation.) (c) Graph A = A(x). Choose the correct graph below. O A. O B. O C. OD. Xmin = 0, Xmax = 3, Xmin = 0, Xmax = 3, Xmin = 0, Xmax = 3, Xmin = 0, Xmax = 3, Xscl = 1, Ymin = 0, Xscl = 1, Ymin = 0, Xscl = 1, Ymin = 0, Xscl = 1, Ymin = 0, Ymax = 15, Yscl = 3 Ymax = 15, Yscl = 3 Ymax = 15, Yscl = 3 Ymax = 15, Yscl = 3 A is largest for x = 1 (1) (Round to two decimal places as needed.) (d) The largest area is ( 2) (Round to two decimal places as needed.) (1) O cubic unit(s). (2) O unit(s). O unit(s). O cubicunit(s). O square unit(s). O square unit(s).3. Two cars are approaching an intersection. One is 5 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 1 mile east of the intersection and is moving at a constant speed of 10 miles per hour. (a) Express the distance d between the cars as a function of time t. (Hint: At t= 0. the cars are 5 miles south and 1 mile east of the intersection, respectively.) (b) Use a graphing utility to graph d = d(t). For what value of t is d smallest? (Round to three decimal places.) 5. An open box with a square base is to be made from a square piece of cardboard 27 inches on a side by cutting out a square from each corner and turning up the sides. X See the figure. Answers parts (a) through (e). 27 in. XV 27 in. (a) Express the volume V of the box as a function of the length x of the side of the square cut from each corner. V = (Simplify your answer.) (b) What is the volume if a 4-inch square is cut out? The volume is ( 1 ) (Simplify your answer.) (c) What is the volume if a 12-inch square is cut out? The volume is ( 2 ) (Simplify your answer.) (d) Graph V= V(x). For what value of x is V largest? Choose the correct graph below. O A. O B. O C. OD. Xmin = 0, Xmax = 13.5, Xmin = 0, Xmax = 13.5, Xmin = 0, Xmax = 13.5, Xmin =0, Xmax = 13.5, Xscl = 0.5, Ymin = 0, Xscl = 0.5, Ymin = 0 Xscl = 0.5, Ymin = 0, Xscl = 0.5, Ymin = 0, Ymax = 1500, Yscl = 250 Ymax = 1500, Yscl = 250 Ymax = 1500, Yscl = 250 Ymax = 800, Yscl = 100 For what value of x is V largest? X= (3) (e) What is the largest volume? The largest volume is (4) (1) O cubic inches. (2) O cubic inches. (3) O cubic inches (4) O inches. inches. O inches. inches O square inches. square inches. O square inches. O square inches O cubic inches.1. Find the average rate of change of f(x) = 2x2 - 3 from 3 to 9. The average rate of change of f(x) = 2x2 - 3 from 3 to 9 is (Simplify your answer.)3. In parts (a) and (b), use the given gure. (a) For what value of x does f(x) = 900? (a) Solve the equation: f(x) = 90:). x = |:| (b) Solve the inequality f(x) > 900. (b) For which values of x is f(x) > 900? For every x in the interval |:|. f(x) > 900. (Type your answer in interval notation.) 4. The cost C. in dollars, to tow a car is modeled by the function 000 = 2.5x+ 85. where x is the number of miles towed. (a) What is the cost of towing a car 40 miles? (b) If the cost of towing a car is $250. how many miles was it towed? (c) Suppose that you have only $150. What is the maximum number of miles that you can be towed? (d) What is the domain of C? (a) The cost is 5:. (Type an integer or a decimal.) (b) The car was towed |:| miles. (Type an integer or a decimal.) (c) The car can be towed a maximum of |:| miles. (Type an integer or a decimal.) (d) Choose the correct answer below. 0 A. (0,00) 0 e. [35.00) 0 C. (- 00,00) 0 D. [0.00) 5. The point at which a company's prots equal zero is called the company's break-even point. Let R represent a company's revenue, let G represent the company's costs, and let 3:: represent the number of units produced and sold each day. R(x) = 200x C(x) = 1 10.5): + 89,500 (a) Find the rm's break-even point; that is, nd x so that R = C. (b) Find the values of x such that R(x) > 000. This represents the number of units that the company must sell to earn a prot. (a) x= |:| (Type a whole number.) (b) Solve the inequality for x. Type the correct inequality symbol in the rst answer box below, and type an integer in the second answer box. x|:||:| 6. Find an equation for the line with the given properties. Express the equation in slope-intercept form. Containing the points P = ( - 3. - 2) and Q = ( - 1,1) What is the equation of the line? y=|:| (Simplify your answer. Use integers or fractions for any numbers in the expression.) 9. The following data represent the weight (in grams) of candy "-2:- bars and the corresponding "m "Is:- "umbmfca'on'es- "-1. -s- (3) Draw a scatter diagram of the data treating weight as the independent variable. Choose the correct scatter diagram below. - 350 x weight x weight x weight (b) What type of relation appears to exist between the weight of a candy bar and the number of calories? 0 Nonlinear 0 Linear (c) Find the equation of the line that passes through (46.371) and (64,434) in slope-intercept form. (Type an expression using x as the variable. Use integers or decimals for any numbers in the expression.) (d) Graph the line on the scatter diagram from part (a). Choose the correct graph below. (9) Use the linear equation you found in part (c) to predict the number of calories in a candy bar that weighs 52.1 grams. |:| cams (f) Interpret the slope of the line found in part (c). If the weight of the candy bar is increased by 1 gram, then the number of calories will increase by |:| calories