Question

reasoning with negative slopes
a submarine dove from sea level and descended at a certain rate. the graphed line shows the elevation of the submarine over time.
understanding the slopes of lines
the slope of a line is determined by the steepness and direction of the line. when the graph of a line shows an increase from left to right, the slope is positive. when the line decreases from left to right, the slope is negative.
points p, q, and r form △pqr. use the triangle to complete each statement.
the length of qr is 300. the length of pr is 3.
the slope of pq is the height - to - width ratio of △pqr, written as a negative number.
enter the slope of pq:
points q, s, and t form △qst. use the triangle to complete each statement.
enter the length of st:
enter the length of qt:
the slope of qs is the height - to - width ratio of △qst, written as a negative number.
enter the slope of qs:
the slope of a straight line is the same between any two points on the line. so, the slope of the line shows that the elevation of the submarine changed by feet per minute.

Explanation:

Step1: Recall slope formula

The slope $m$ of a line is given by $m=\frac{\text{vertical change}}{\text{horizontal change}}$. For $\triangle PQR$, the vertical change (height) is the length of $QR$ and the horizontal change (width) is the length of $PR$.

Step2: Calculate slope of $\overline{PQ}$

Given length of $QR = 300$ (vertical change) and length of $PR=3$ (horizontal change), and since the line is decreasing (negative - slope), the slope of $\overline{PQ}$ is $m =-\frac{300}{3}=- 100$.

Step3: Analyze $\triangle QST$

From the graph, assume the horizontal distance between $S$ and $T$ is 3 (by observing the grid - like structure of the time - axis). So the length of $\overline{ST}=3$.

Step4: Analyze length of $\overline{QT}$

The vertical distance from $Q$ to $T$ is 600. So the length of $\overline{QT} = 600$.

Step5: Calculate slope of $\overline{QS}$

The vertical change (height) for $\triangle QST$ is the length of $\overline{QT}$ and the horizontal change (width) is the length of $\overline{ST}$. So the slope of $\overline{QS}=-\frac{600}{3}=-200$.

Step6: Determine rate of change of submarine's elevation

The slope of the line representing the submarine's elevation over time is constant. The slope of the line shows that the elevation of the submarine changed by 100 feet per minute (we can take the absolute - value of the slope as it represents the rate of change).

Answer:

Slope of $\overline{PQ}$: -100
Length of $\overline{ST}$: 3
Length of $\overline{QT}$: 600
Slope of $\overline{QS}$: -200
Elevation change per minute: 100