1a The pH of a substance measures its acidity or alkalinity

1a. The pH of a substance measures its acidity or alkalinity. It is found by the formula

pH = log[H⁺]

where [H⁺] is the concentration of hydrogen ions in an aqueous solution given in moles per liter. Find the pH (to the nearest tenth) for the solution.

toothpaste with [H⁺] = 1.15 × 10⁹

1b.An advertising agency conducted a survey and found that the number of units sold, N, is related to the amount a spent on advertising (in dollars) by the following formula:

N = 4,500 + 100 ln a    (a 1)

How many units are sold after spending $60,000

1c. The \"learning curve\" describes the rate at which a person learns certain tasks. If a person sets a goal of typing N words per minute (wpm), the length of time t (in days) to achieve this goal is given by

t = 62.5 ln

(1 n/80)

(1) According to this formula, what is the maximum number of words per minute

(2) Solve for N

Solution

1. pH = -log(H⁺)
pH = -log(1.15x10^-9)
pH = -(log1.15 + 9log10)
pH = -(log1.15 + 9)
pH = 8.93

2. N = 4500 + 100 lna
N = 4500 + 100 ln60000
N = 4500 + 100(11)
N = 5600 units

3. t = -62.5 ln(1-n/80)
1 = -62.5 ln(1-n/80
-1/62.5 = ln(1-n/80)
-0.016 = ln(1-n/80)
exp(-0.016) = 1-n/80
0.9841 = 1-n/80
-0.0159 = -n/80
n = 80*0.0159
n = 1.272 wpd

n = 1.272 days * 24 h/days * 3600 s/1 h = 109901 word/min