lesson 4 assignment continued
3 dudley and bob monitored the distance their pet turtle could walk in a certain amount of time. the table shows their results. the table of values represents a proportional relationship.
(a) define variables for the quantities that are changing in this problem situation.
(b) write an equation for the distance traveled by the turtle based on the number of minutes.
(c) identify the constant of proportionality in part (b). what does it mean in this context?
(d) use your equation in part (b) to write an equation for the time it would take for the turtle to travel a given distance.
(e) identify the constant of proportionality in part (d). what does it mean in this context?
stretch optional
if a varies inversely as b, then the product of a and b is constant.
1 use direct and inverse variation to solve this problem.
if x varies inversely as y and varies directly with z, and if y and z are both 3 when x = 5, what is the value of z - y when x = 8?

Question

Accepted Answer

### Part a
# Brief Explanations:
Let $x$ represent the time (in minutes) and $y$ represent the distance (in inches). These are the two quantities that change as per the problem - time for which the turtle walks and the distance it covers.
# Answer:
Let $x$ be the time (in minutes) and $y$ be the distance (in inches).

### Part b
# Explanation:
## Step1: Find the constant of proportionality $k$
For a proportional relationship $y = kx$. Using the first pair $(x = 5,y=14.5)$, we find $k=\frac{y}{x}$. So $k=\frac{14.5}{5}=2.9$
## Step2: Write the equation
Since $k = 2.9$, the equation for the distance $y$ in terms of time $x$ is $y=2.9x$
# Answer:
$y = 2.9x$

### Part c
# Brief Explanations:
The constant of proportionality $k = 2.9$. In the context of the problem, since $y$ (distance) and $x$ (time) are related by $y=kx$, $k$ represents the speed of the turtle. The turtle walks at a speed of $2.9$ inches per minute.
# Answer:
The constant of proportionality is $2.9$. It means the turtle's speed is $2.9$ inches per minute.

### Part d
# Explanation:
## Step1: Start with the equation $y = 2.9x$
We want to solve for $x$ in terms of $y$. Divide both sides of the equation $y = 2.9x$ by $2.9$.
## Step2: Derive the new equation
$x=\frac{y}{2.9}$
# Answer:
$x=\frac{y}{2.9}$

### Part e
# Brief Explanations:
The constant of proportionality in $x=\frac{y}{2.9}$ is $\frac{1}{2.9}\approx0.345$. In this context, since $x$ (time) and $y$ (distance) are related by $x=\frac{y}{2.9}$, it represents the time taken by the turtle to cover one - inch distance.
# Answer:
The constant of proportionality is $\frac{1}{2.9}\approx0.345$. It means the turtle takes approximately $0.345$ minutes to walk one inch.

### Stretch Problem
# Explanation:
## Step1: Set up the equations based on direct and inverse variation
Since $x$ varies inversely as $y$ and directly as $z$, we have $x=\frac{kz}{y}$. When $x = 5,y = 3,z = 3$, substitute into the equation $5=\frac{k\times3}{3}$, so $k = 5$. The equation is $x=\frac{5z}{y}$
## Step2: Find $z$ and $y$ when $x = 8$
Let $y=m$ and $z=n$. The equation is $8=\frac{5n}{m}$, or $n=\frac{8m}{5}$. But we can also note that from the original relationship (since the relationship between $x,y,z$ is fixed by $x=\frac{5z}{y}$), when $x = 8$, we can assume $y$ and $z$ still follow the same proportionality. Cross - multiplying gives $8y=5z$. Let's assume $y = 5t$ and $z = 8t$ (to satisfy $8y = 5z$). Substituting back into the original equation $x=\frac{5z}{y}$, when $x = 8$, it holds.
## Step3: Calculate $z - y$
$z-y=8t - 5t=3t$. From the original values ($x = 5,y = 3,z = 3$) and the equation $x=\frac{5z}{y}$, when $x = 8$, if we use $x=\frac{5z}{y}$, and we know from the proportion $z=\frac{8}{5}y$. Also, since the relationship is consistent, we can use the fact that $x=\frac{5z}{y}$. When $x = 8$, $8=\frac{5z}{y}$, or $z=\frac{8y}{5}$. If we assume $y = 5$ (to make the calculation simple, because of the ratio $z:y = 8:5$), then $z = 8$
$z - y=8 - 5=3$
# Answer:
$3$

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QUESTION IMAGE

Question

Explanation:

Part a

Step1: Find the constant of proportionality \(k\)

Step2: Write the equation

Answer:

Part b