Question

graphical analysis

1. the following times were measured for spheres of different masses to be pushed a distance of 1.5 meters by a stream:

mass (kg) time (s)
5 10.2
10 17.3
15 23.8
20 31.0
a. graph the data by hand on the grid provided and write a mathematical model for the graph that describes the data.
b. write a clear sentence that describes the relationship between mass and time.

1. a student performed an experiment with a metal sphere. the student shot the sphere from a slingshot and measured its maximum height. the sphere was shot six times at six different angles above the horizon.

a. what is the relationship being studied?
b. what is the independent variable in this experiment?
c. what is the dependent variable in this experiment?
d. what variables must be held constant throughout this experiment?

1. a. write a mathematical model that describes the graph.

b. provide an interpretation for the y - intercept.
c. using the mathematical model, how many applications would be needed to earn $8000?

Explanation:

1.

a.

Step1: Assume the relationship is linear of the form $y = mx + b$, where $y$ is time $t$, $x$ is mass $m$.

We use the least - squares method or find the slope $m$ and y - intercept $b$ by taking two points. Let's take $(m_1,t_1)=(5,10.2)$ and $(m_2,t_2)=(10,17.3)$.
The slope $m=\frac{t_2 - t_1}{m_2 - m_1}=\frac{17.3 - 10.2}{10 - 5}=\frac{7.1}{5}=1.42$.
Using the point - slope form $t - t_1=m(m - m_1)$ with $(m_1,t_1)=(5,10.2)$ and $m = 1.42$, we get $t-10.2 = 1.42(m - 5)$.
Expanding gives $t=1.42m+3.1$.

As the mass of the sphere increases, the time it takes for the sphere to be pushed a distance of 1.5 meters by the stream increases linearly.

The student is studying the relationship between the angle at which the metal sphere is shot from the slingshot and the maximum height it reaches.

Answer:

The mathematical model is $t = 1.42m+3.1$.

b.