Question

a pizza is taken out of an oven and placed on a counter. the temperature, t, in degrees fahrenheit, of the pizza after m minutes is modeled by the function t = 72 + 200e^{-0.045m}. which graph represents the model? (graph with y-axis labeled °f from 0 to 400, x-axis labeled minutes from 0 to 50, and a curve starting at (0, 200) and decreasing towards 72 as minutes increase)

Explanation:

Step1: Analyze the function at \( m = 0 \)

Substitute \( m = 0 \) into \( T = 72 + 200e^{-0.045m} \). Since \( e^0 = 1 \), we get \( T = 72 + 200(1) = 272 \). Wait, but the graph shown has a y - intercept around 200? Wait, maybe a typo in the function? Wait, maybe the function is \( T = 72+ 200e^{-0.045m} \), but when \( m = 0 \), \( T = 272 \). But the given graph has a y - intercept of 200. Wait, maybe the function is \( T=72 + 128e^{-0.045m} \)? No, the problem says \( T = 72+200e^{-0.045m} \). Alternatively, maybe the graph is of a cooling function. The general form of a cooling function is \( T=T_s+(T_0 - T_s)e^{-kt} \), where \( T_s \) is the surrounding temperature (here 72), \( T_0 \) is the initial temperature. So initial temperature \( T_0=72 + 200=272 \). The function is a decreasing exponential function (since the exponent has a negative sign) that approaches 72 as \( m\to\infty \).

Step2: Analyze the shape and asymptote

The function \( T = 72+200e^{-0.045m} \) is an exponential decay function shifted up by 72. So as \( m \) increases, \( T \) approaches 72. The graph should start at \( T = 272 \) when \( m = 0 \) and decrease towards 72. The given graph (the one with the curve starting around 200? Wait, maybe there was a typo in the function, but assuming the graph shown is a cooling curve (decreasing exponential, approaching a horizontal asymptote). Let's check the asymptote: as \( m\to\infty \), \( e^{-0.045m}\to0 \), so \( T\to72 \). The graph shown has a horizontal asymptote? The curve is decreasing and approaching a value (maybe 72, but the y - axis is labeled with 0,100,200,300,400). So the graph is a decreasing exponential function, which matches the form of the cooling function \( T = 72+200e^{-0.045m} \) (even if the initial value seems off, the shape is an exponential decay, starting from a higher value and approaching 72). So the graph shown (the one with the curve starting at around 200? Wait, maybe the function was supposed to be \( T = 72+128e^{-0.045m} \), but regardless, the key is the shape: exponential decay, so the graph with a decreasing curve that approaches a horizontal line (the asymptote at \( T = 72 \)) is the correct one. The given graph (the one in the image) is a decreasing exponential curve, so it represents the cooling model.

Answer:

The graph (the one shown with the curve starting from the left - hand side, decreasing, and approaching a horizontal asymptote) represents the model \( T = 72 + 200e^{-0.045m} \) as it is a decreasing exponential function (cooling curve) which matches the form of the given temperature - time model.