Section 7.5

Formulas, Applications and Variation

Direct Variation

Direct Variation When a situation is modeled by a linear function of

the form f(x) = kx or y = kx,

where k is a nonzero constant, we say that there is direct variation that y varies directly as x that y is proportional to x

The number k is called the variation constant, or the constant of proportionality.

Direct Variation

Find k if y varies directly as x given y=30 when x=5, then find the equation y varies directly as x use the direct variation formula y = kx 30 = k(5) → 6 = k

Equation → y = 6x

Direct Variation

The number of calories burned while dancing is directly proportional to the time spent. It takes 25 minutes to burn 110 calories, how long would it take to burn 176 calories when dancing.

Direct Variation

The number of calories burned while dancing is directly proportional to the time spent. It takes 25 minutes to burn 110 calories, how long would it take to burn 176 calories when dancing.

y = kx It takes 25 minutes to burn 110 calories

110ca = k(25min) → 4.4ca/min = k

how long would it take to burn 176 calories 176ca = (4.4ca/min)(x) → 40min = x

It will take 40 minutes to burn 176 calories

Inverse Variation

Inverse Variation When a situation is modeled by a rational function

of the form f(x) = k/x or y = k/x,

where k is a nonzero constant, we say that there is inverse variation that y varies inversely as x that y is inversely proportional to x

The number k is called the variation constant, or the constant of proportionality.

Inverse Variation

Find k if y varies inversely as x given y = 27 when x = 1/3, then find the equation y varies inversely as x = inverse variation formula y = k/x 27 = k/(1/3) → 9 = k

Equation → y = 9/x

Inverse Variation

The frequency of a string is inversely proportional to its length. A violin string that is 33 cm long vibrates with a frequency of 260Hz. What is the frequency when the string is shortened to 30cm?

Inverse Variation

The frequency of a string is inversely proportional to its length. A violin string that is 33 cm long vibrates with a frequency of 260Hz. What is the frequency when the string is shortened to 30cm?

y = k / x A violin string that is 33 cm long vibrates with a

frequency of 260Hz. 260Hz = k / 33cm → 8580HZ/cm = k

What is the frequency when the string is shortened to 30cm y = 8580HZ/cm / 30cm → y = 286Hz

The frequency of the string is 286 Hz

Joint Variation

Joint Variation Y varies jointly as x and z if, for some nonzero

constant k, y = kxz

Joint Variation

The drag Force F on a boat varies jointly as the wetted surface area A and the square of the velocity of the boat. If the boat traveling 6.5 mph experiences a drag force of 86N when the wetted surface area is 41.2ft² find the wetted surface area of a boat traveling 8.2mph with a drag force of 94N

Joint Variation

The drag Force F on a boat varies jointly as the wetted surface area A and the square of the velocity of the boat. If the boat traveling 6.5 mph experiences a drag force of 86N when the wetted surface area is 41.2ft² find the wetted surface area of a boat traveling 8.2mph with a drag force of 94N

y = kxz² 86N = k(41.2ft²)(6.5mph)² →k = .049405 N/ft²mph²

N (Newtons) = kg m/s2⋅ 94N = 0.049(x)(8.2mph)² →x = 28.530ft² The wetted surface area is 28.53 ft²

Homework

Section 7.5 44, 50, 57, 59, 69, 71, 73, 75, 77, 80