The length of a spring is a linear function of the mass of the object hanging from it. For a particular spring, the length was 12 centimeters when a 2-kg mass was attached to it. When a 5-kg mass was hung from it, it stretched to a length of 18 cm. How much mass must be hung from this spring for it to stretch to 25 cm?

d = k x + b by the way b is un stretched length

12 = 2 k + b
18 = 5 k + b
----------------- subtract
-6 = -3 k
k = 2 then b = 8
25 = 2 m + 8

To solve this problem, we can use the concept of linear functions and proportions.

Let's start by finding the rate of change, or the slope of the linear function. We can calculate the change in length divided by the change in mass:

Slope = (Change in length) / (Change in mass)
= (18 cm - 12 cm) / (5 kg - 2 kg)
= 6 cm / 3 kg
= 2 cm/kg

Now that we have the slope of the linear function, we can write an equation using the point-slope form:

y - y1 = m(x - x1)

Here, the x-coordinate represents the mass, the y-coordinate represents the length, and (x1, y1) is one of the given points (2 kg, 12 cm). Plugging in these values:

y - 12 = 2(x - 2)

Next, we need to find the mass (x) that corresponds to a length (y) of 25 cm. We'll plug in 25 for y and solve for x:

25 - 12 = 2(x - 2)
13 = 2(x - 2)
13 = 2x - 4
17 = 2x
x = 17 / 2
x = 8.5 kg

Therefore, you would need to hang a mass of 8.5 kg from this spring for it to stretch to a length of 25 cm.

To find the amount of mass that needs to be hung from the spring in order for it to stretch to 25 cm, we can first determine the rate of change of the length with respect to the mass.

Given that the length of the spring is a linear function of the mass, we can use the formula for a linear function:

y = mx + b

where:
y = Length of the spring (in cm)
m = Slope of the linear function (rate of change)
x = Mass of the object (in kg)
b = y-intercept (length of the spring when no mass is attached)

From the given information, we have two points on the linear function:
(2 kg, 12 cm) and (5 kg, 18 cm).

Using the formula for slope (m) between two points:

m = (change in y) / (change in x)

m = (18 cm - 12 cm) / (5 kg - 2 kg)
m = 6 cm / 3 kg
m = 2 cm/kg

Now that we know the slope of the linear function, we can determine the y-intercept (b) by substituting one of the points into the equation:

12 cm = (2 cm/kg)(2 kg) + b
12 cm = 4 cm + b
b = 12 cm - 4 cm
b = 8 cm

The equation for the length of the spring is now:

y = 2 cm/kg * x + 8 cm

To find the mass required for a length of 25 cm, we substitute y = 25 cm into the equation and solve for x:

25 cm = 2 cm/kg * x + 8 cm
17 cm = 2 cm/kg * x
x = 17 cm / (2 cm/kg)
x = 8.5 kg

Therefore, in order for the spring to stretch to 25 cm, a mass of 8.5 kg must be hung from it.