MATH 1001: Pre-Calculus Mathematics 3 14) Find an equation of a rational function f that satisfies the given conditions: |4 marks] Vertical asymptotes: x = -5,x = 2 Horizontal asymptotes: y = -2 x-intercepts: x = -6,x = 4 f(-2) = -4 Section 3.6 15) Page 241, Q. 10 [2 marks] 16) Page 241, Q. 34 [3 marks]3.6 Variation 241 W = 125 Wid- Guideline 4 To answer the question, we substitute w = 2, d = 8, and / = 10 into the formula found in guideline 3, obtaining W = 125 . 2 . 8' 10 - = 1600 lb. 3.6 Exercises Exer. 1-16: Express the statement as a formula that involves 12 r is directly proportional to the product of s and v and the given variables and a constant of proportion- inversely proportional to the cube of p. If s = 2, v = 3, and ality k, and then determine the value of & from the given p = 5, then r = 40. conditions. r - kik - 250 1 a is directly proportional to v. If v = 30, then a = 12. 13 q is inversely proportional to the sum of x and y. If x = 0.5 and y = 0.7, then q = 1.4. 2 s varies directly as f. If r = 10, then s = 18. s - kr, x - ! * : x - 1.68 14 y is directly proportional to x and inversely proportional to 3 V varies directly as the cube of r. If r = 3, then V = 36r. the sum of rand s. If x = 3, F = 5, and s = 7, then y = 2. Y - K . K - S 4 8 is directly proportional to the square of x. If x = 2, then 15 y is directly proportional to the square root of x and $ = 24. 5 - Wr k - 6 inversely proportional to the cube of z. If x = 9 and z = 2, then y = 5. 5 p varies directly as s and inversely as t. If s = -2 and t = 4, then r = 7. 16 y is directly proportional to the square of x and inversely r - k=;k- -14 proportional to the square root of z. If x = 5 and z = 16, then y = 10. 6 w varies directly as z and inversely as the square root of a. If z = 2 and u = 9, then w = 6. W-;x-9 17 Liquid pressure The pressure P acting at a point in a liquid is directly proportional to the distance d from the sur- 7 y is directly proportional to the square of x and inversely proportional to the cube of z. If x = 5 and z = 3, then face of the liquid to the point. y = 25. (a) Express P as a function of d by means of a formula that y- KJ; k - 27 involves a constant of proportionality &. P - kd 8 y is directly proportional to x and inversely proportional to (b) In a certain oil tank, the pressure at a depth of 2 feet is the square of z. If x = 4 and z = 3, then y = 16. 1 18 1b/fr. Find the value of & in part (a). y- kj:x- 36 (c) Find the pressure at a depth of 5 feet for the oil tank 9 z is directly proportional to the product of the square of x in part (b). 295 Thyn] and the cube of y. If x = 7 and y = -2, then z = 16. z - krlyh k - - 7 (d) Sketch a graph of the relationship between P and d for 10 z is directly proportional to the product of x and the cube root of y. If x = 2 and y = 8, then z = 12. 18 Hooke's law Hooke's law states that the force F required to stretch a spring & units beyond its natural length is directly proportional to x. 11 z is directly proportional to the product of x and y and inversely proportional to the cube root of w. If x = 6, y = 4, (a) Express F as a function of x by means of a formula that and w = 27, then z = 16. involves a constant of proportionality &. F - kx VW