Ex 1: If a linear function satisfies the conditions of h( 3) = 1 and h(3) = 2, find h(x).
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1 In lesson 1, the definition of a linear function was given. A linear function is a function of the form f(x) = ax + b, where a is the slope of the line and (0, b) is the y-intercept. A linear function can be determined if given the slope and a point (ordered pair) or two points (two ordered pairs). Ex 1: If a linear function satisfies the conditions of h( 3) = 1 and h(3) = 2, find h(x). Ex 2: A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. The radius r is yet to be determined. (a) Express the surface area A as a function of the radius r. Note: Read directions in LON CAPA carefully. If the questions asks for a function, be certain to use function notation such as A(r) = rather than just =. (b) If r = 2 feet, find the surface area of the tank. Cross-section of tank Cross-section of tank πr 2 Circle area = Area of Cylinder = 2πrh (tube part only) Surface Area = area of cylinder (no top or bottom) + area of sphere 1
2 Ex 3: For children between ages 6 and 10 (inclusively), height y (in inches) is frequently a linear function of age t (in years). The height of a certain child is 48 inches at age 6 and 50.5 inches at age 7. a) What is the slope? What does it represent? (You will need to find two points of the form (age, height). b) Express the height of the child as a function of years. c) Use your function from part (a) to predict the height of the child at age 10. Ex 4: See the figure below. Triangle ABC is inscribed in a semicircle of diameter 15. a) If x denotes the length of side AC, express the length of y (side BC) as a function of x. (A triangle inscribed in a semicircle will be a right triangle. Therefore, angle ACB will be a right angle.) b) Express the area of triangle ABC as a function of x. State the domain of this function. C x y A 15 B 2
3 a) Relate the 3 sides of the right triangle to get y as a function of x. b) A = 1 bh (Let x be base b and y be height h.) 2 Use a sign chart to find the domain of function A. 3
4 Ex 5: An open-top cylindrical can is to have a volume of 60π cubic inches. (See picture below.) Volume of a cylinder = πr 2 h Surface area of a cylinder = Area of circular top and bottom plus are of the tube. Area of the tube = circumference of circular base time the height a) Express the height h of the can as a function of the radius. b) Express the surface area of the can as a function of the radius. (Hint: Surface area is the sum of the area of the circular bottom plus a rectangle that has a length that is the circumference of the circular bottom and a width that is the height of the can.) c) Find the surface area of the can if the radius is 3 inches. Round to hundredths place. a) b) Remember: There is no top to this cylinder. 4
5 Ex 6: The area enclosed by a circle, in square feet, is a function of its radius r in feet. This relation is expressed as A(r) = pi r 2. (a) Find A(14) in terms of pi. (b) If A(r) = 529π, determine the value of r, the radius. Ex 7: The volume V of enclosed by a certain cylinder, in cubic inches, is a function of its radius r in inches. This relation is expressed by the function V(r) = 10π r 2. (a) Find V(21). Express answer using pi. (b) If V(r) = 1690π, find the value of r, the radius. Ex 8: The temperature T in degrees Celsius t hours after 5 AM is given by the function below. T(t) = 1 3 t2 + 4t + 1 for 0 t 10 Find T(0) and T(10). 5
6 Ex 9: The function C(x) = x 2 x + 40 models the cost, in hundreds of dollars, to produce x thousand items. Find the total cost to produce 26,000 items and 491,000 items. Ex 10: 125 The population of rabbits in a garden area can be modeled by the function Pt ( ), t 5 where t represent the number of years since Find the rabbit population in 2010 and
7 Ex 11: A box is to be created from a piece of cardstock measuring 16 inches by 25 inches by cutting out identical squares with side lengths of x from each corner and folding up the sides. Express the volume V of the box as a function of x. Ex 12: A farmer wishes to create a pen with an area of 2500 square feet with a fence in the middle dividing the pen into two areas. He will put gates on one side and on the middle side. The gates will have a width of 5 feet. (See the picture.) (a) Express the length of the pen L as a function of the width w. (b) If the fencing cost $8 per foot express the cost of the fencing C function of the width w. Do NOT include the lengths of the gates. 7
221 MATH REFRESHER session 3 SAT2015_P06.indd 221 4/23/14 11:39 AM
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