Question

a lake near the arctic circle is covered by a 2-meter-thick sheet of ice during the cold winter months. when spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. after 3 weeks, the sheet is only 1.25 meters thick. let ( y ) represent the ice sheet’s thickness (in meters) after ( x ) weeks. which of the following information about the graph of the relationship is given? choose 1 answer.
a slope and ( x )-intercept
b slope and ( y )-intercept
c slope and a point that is not an intercept
d ( x )-intercept and ( y )-intercept
e ( y )-intercept and a point that is not an intercept
f two points that are not intercepts

Explanation:

Step1: Analyze the problem context

We have a situation where the ice thickness \( y \) (in meters) after \( x \) weeks is being modeled. Initially (when \( x = 0 \), at the start of spring), the ice thickness is 2 meters. So the \( y \)-intercept (when \( x = 0 \)) is 2. After 3 weeks (\( x = 3 \)), the thickness is 1.25 meters. We can find the slope (rate of change) using the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(0,2) \) and \( (x_2,y_2)=(3,1.25) \), then \( m=\frac{1.25 - 2}{3 - 0}=\frac{- 0.75}{3}=- 0.25 \). So we have the slope (rate of change) and the \( y \)-intercept (initial thickness when \( x = 0 \)).

Step2: Evaluate the options

• Option A: Slope and \( x \)-intercept. The \( x \)-intercept is when \( y = 0 \), we haven't calculated that directly from the given info (we have initial \( y \) when \( x = 0 \), not \( x \) when \( y = 0 \) from the given data).
• Option B: Slope and \( y \)-intercept. We found the slope (rate of decrease) and the \( y \)-intercept (initial thickness at \( x = 0 \)) from the given information (initial thickness 2 meters, and thickness after 3 weeks to find slope).
• Option C: Slope and a point that is not an intercept. We have the \( y \)-intercept as a point \( (0,2) \), so this is not correct.
• Option D: \( x \)-intercept and \( y \)-intercept. We don't have the \( x \)-intercept from the given info (we have \( y \)-intercept and a point to find slope).
• Option E: \( y \)-intercept and a point that is not an intercept. The point \( (3,1.25) \) is used to find slope, but we can also think in terms of having slope (from two points including \( y \)-intercept) and \( y \)-intercept. But the key is we have slope (rate) and \( y \)-intercept (initial value).
• Option F: Two points that are not intercepts. We have \( (0,2) \) which is a \( y \)-intercept, so this is not correct.

Answer:

B. Slope and \( y \)-intercept