Earned run average A baseball pitcher’s earned run average (ERA) is A ( e , i ) = 9 e / i , where e is the number of earned runs given up by the pitcher and i is the number of innings pitched. Good pitchers have low ERAs. Assume that e ≥ 0 and i > 0 are real numbers. a. The single-season major league record for the lowest ERA was set by Dutch Leonard of the Detroit Tigers in 1914. During that season, Dutch pitched a total of 224 innings and gave up just 24 earned runs. What was his ERA? b. Determine the ERA of a relief pitcher who gives up 4 earned runs in one-third of an inning. c. Graph the level curve A ( e , i ) = 3 and describe the relationship between e and i in this case.
Earned run average A baseball pitcher’s earned run average (ERA) is A ( e , i ) = 9 e / i , where e is the number of earned runs given up by the pitcher and i is the number of innings pitched. Good pitchers have low ERAs. Assume that e ≥ 0 and i > 0 are real numbers. a. The single-season major league record for the lowest ERA was set by Dutch Leonard of the Detroit Tigers in 1914. During that season, Dutch pitched a total of 224 innings and gave up just 24 earned runs. What was his ERA? b. Determine the ERA of a relief pitcher who gives up 4 earned runs in one-third of an inning. c. Graph the level curve A ( e , i ) = 3 and describe the relationship between e and i in this case.
Solution Summary: The author explains that Dutch Leonard's earned run average A is 0.9643. The relief pitcher gave up 4 earned runs in one-third of an innings.
Earned run average A baseball pitcher’s earned run average (ERA) is A(e, i) = 9e/i, where e is the number of earned runs given up by the pitcher and i is the number of innings pitched. Good pitchers have low ERAs. Assume that e ≥ 0 and i > 0 are real numbers.
a. The single-season major league record for the lowest ERA was set by Dutch Leonard of the Detroit Tigers in 1914. During that season, Dutch pitched a total of 224 innings and gave up just 24 earned runs. What was his ERA?
b. Determine the ERA of a relief pitcher who gives up 4 earned runs in one-third of an inning.
c. Graph the level curve A(e, i) = 3 and describe the relationship between e and i in this case.
A body of mass m at the top of a 100 m high tower is thrown vertically upward with an initial velocity of 10 m/s. Assume that the air resistance FD acting on the body is proportional to the velocity V, so that FD=kV. Taking g = 9.75 m/s2 and k/m = 5 s, determine: a) what height the body will reach at the top of the tower, b) how long it will take the body to touch the ground, and c) the velocity of the body when it touches the ground.
A chemical reaction involving the interaction of two substances A and B to form a new compound X is called a second order reaction. In such cases it is observed that the rate of reaction (or the rate at which the new compound is formed) is proportional to the product of the remaining amounts of the two original substances. If a molecule of A and a molecule of B combine to form a molecule of X (i.e., the reaction equation is A + B ⮕ X), then the differential equation describing this specific reaction can be expressed as:
dx/dt = k(a-x)(b-x)
where k is a positive constant, a and b are the initial concentrations of the reactants A and B, respectively, and x(t) is the concentration of the new compound at any time t. Assuming that no amount of compound X is present at the start, obtain a relationship for x(t). What happens when t ⮕∞?
Consider a body of mass m dropped from rest at t = 0. The body falls under the influence of gravity, and the air resistance FD opposing the motion is assumed to be proportional to the square of the velocity, so that FD = kV2. Call x the vertical distance and take the positive direction of the x-axis downward, with origin at the initial position of the body. Obtain relationships for the velocity and position of the body as a function of time t.
Chapter 15 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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