World Military Expenditure The following chart shows total military and arms trade expenditure from 2011–2020 (t = 1 represents 2011). †A bar graph titled "World military expenditure" has a horizontal t-axis labeled "Year since 2010" and a vertical axis labeled "$ (billions)". The bar graph has 10 bars. Each bar is associated with a label and an approximate value as listed below. 1: 1,800 billion dollars 2: 1,775 billion dollars 3: 1,750 billion dollars 4: 1,730 billion dollars 5: 1,760 billion dollars 6: 1,760 billion dollars 7: 1,850 billion dollars 8: 1,900 billion dollars 9: 1,950 billion dollars 10: 1,980 billion dollars (a) If you want to model the expenditure figures with a function of the form f(t) = at2 + bt + c, would you expect the coefficient a to be positive or negative? Why? HINT [See "Features of a Parabola" in this section.] We would expect the coefficient to be positive because the curve is concave up. We would expect the coefficient to be negative because the curve is concave down. We would expect the coefficient to be negative because the curve is concave up. We would expect the coefficient to be positive because the curve is concave down. (b) Which of the following models best approximates the data given? (Try to answer this without actually computing values.) f(t) = −6t2 − 44t + 1,840 f(t) = −6t2 + 44t + 1,840 f(t) = 6t2 − 44t − 1,840 f(t) = 6t2 − 44t + 1,840 (c) What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction? Extrapolating in the positive direction leads one to predict more and more steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts more and more steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict less and less steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts more and more steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict more and more steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts less and less steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict less and less steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts less and less steeply increasing military expenditure as we go back in time, contradicting history.

World Military Expenditure The following chart shows total military and arms trade expenditure from 2011–2020 (t = 1 represents 2011). †A bar graph titled "World military expenditure" has a horizontal t-axis labeled "Year since 2010" and a vertical axis labeled "$ (billions)". The bar graph has 10 bars. Each bar is associated with a label and an approximate value as listed below. 1: 1,800 billion dollars 2: 1,775 billion dollars 3: 1,750 billion dollars 4: 1,730 billion dollars 5: 1,760 billion dollars 6: 1,760 billion dollars 7: 1,850 billion dollars 8: 1,900 billion dollars 9: 1,950 billion dollars 10: 1,980 billion dollars (a) If you want to model the expenditure figures with a function of the form f(t) = at2 + bt + c, would you expect the coefficient a to be positive or negative? Why? HINT [See "Features of a Parabola" in this section.] We would expect the coefficient to be positive because the curve is concave up. We would expect the coefficient to be negative because the curve is concave down. We would expect the coefficient to be negative because the curve is concave up. We would expect the coefficient to be positive because the curve is concave down. (b) Which of the following models best approximates the data given? (Try to answer this without actually computing values.) f(t) = −6t2 − 44t + 1,840 f(t) = −6t2 + 44t + 1,840 f(t) = 6t2 − 44t − 1,840 f(t) = 6t2 − 44t + 1,840 (c) What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction? Extrapolating in the positive direction leads one to predict more and more steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts more and more steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict less and less steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts more and more steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict more and more steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts less and less steeply increasing military expenditure as we go back in time, contradicting history. Extrapolating in the positive direction leads one to predict less and less steeply rising military expenditure, which may or may not occur. Extrapolating in the negative direction predicts less and less steeply increasing military expenditure as we go back in time, contradicting history.