Question

the slope of the blue curve measures the planes rate of ascent. the unit of measurement for the slope of the curve is thousands of feet per minute. at point a, the slope of the curve is, which means that the plane is at a rate of feet per minute. (hint: calculating the slope, pay extra attention to the units of analysis.) at point b, the slope of the blue curve is, which means that the plane is at a rate of feet per minute. (hint: calculating the slope, pay extra attention to the units of analysis.)

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{\Delta y}{\Delta x}$. Here, $y$ represents altitude (in thousands of feet) and $x$ represents time (in minutes).

Step2: Estimate coordinates for point A

Suppose point A has coordinates approximately $(6, 30)$ (from the graph). To find the slope at point A, we can consider a small interval around it. Let's assume another point close to A, say $(5, 25)$. Then $\Delta y = 30 - 25=5$ (in thousands of feet) and $\Delta x=6 - 5 = 1$ (in minutes). The slope $m_A=\frac{5}{1}=5$ (thousands of feet per minute). In feet per minute, it is $5\times1000 = 5000$ feet per minute. The plane is ascending at a rate of 5000 feet per minute.

Step3: Estimate coordinates for point B

Suppose point B has coordinates approximately $(2, 5)$. Consider a point close to B, say $(1, 2.5)$. Then $\Delta y=5 - 2.5 = 2.5$ (in thousands of feet) and $\Delta x=2 - 1=1$ (in minutes). The slope $m_B=\frac{2.5}{1}=2.5$ (thousands of feet per minute). In feet per minute, it is $2.5\times1000=2500$ feet per minute. The plane is ascending at a rate of 2500 feet per minute.

Answer:

At point A, the slope of the curve is 5 (thousands of feet per minute), which means that the plane is ascending at a rate of 5000 feet per minute.
At point B, the slope of the blue curve is 2.5 (thousands of feet per minute), which means that the plane is ascending at a rate of 2500 feet per minute.