Transcript 7.5
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
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)
Equation
→
→
9=k
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
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
Homework
Section 7.5
44, 50, 57, 59, 69, 71, 73, 75, 77, 80