Question

the table gives methane gas emissions in millions of metric tons. the quadratic model y = 0.0429x² - 9.73x + 716 approximates the emissions for these years. in the model, x represents the number of years since 2008, so x = 0 represents 2008, x = 1 represents 2009, and so on. complete parts a and b.
a. according to the model, what would emissions be in 2016?
according to the model, in 2016 the emissions would be 640.9 million metric tons.
(round to the nearest tenth as needed.)
b. find the nearest year beyond 2008 for which this model predicts that emissions will reach 500 million metric tons.
the nearest year beyond 2008 for which the model predicts that the emissions will reach 500 million metric tons is
(round down to the nearest year.)

Explanation:

Step1: Determine the value of x for 2016

Since x = 0 represents 2008, for 2016, x=2016 - 2008=8.

Step2: Substitute x into the quadratic - model

Substitute x = 8 into y = 0.0429x²-9.73x + 716.
y=0.0429×8²-9.73×8 + 716
y=0.0429×64-77.84 + 716
y = 2.7456-77.84 + 716
y=640.9056≈640.9

Step3: Set up the equation for part b

Set y = 500 in the equation y = 0.0429x²-9.73x + 716.
0.0429x²-9.73x + 716 - 500=0
0.0429x²-9.73x + 216 = 0

Step4: Use the quadratic formula

The quadratic formula for ax²+bx + c = 0 is x=\frac{-b\pm\sqrt{b² - 4ac}}{2a}. Here, a = 0.0429, b=-9.73, c = 216.
First, calculate the discriminant Δ=b² - 4ac=(-9.73)²-4×0.0429×216
Δ = 94.6729-37.0176=57.6553
x=\frac{9.73\pm\sqrt{57.6553}}{2×0.0429}=\frac{9.73\pm7.6}{0.0858}

Step5: Find the two solutions for x

x1=\frac{9.73 + 7.6}{0.0858}=\frac{17.33}{0.0858}\approx202
x2=\frac{9.73 - 7.6}{0.0858}=\frac{2.13}{0.0858}\approx25

We take the smaller non - negative solution x≈25.

Step6: Find the year

Since x represents the number of years since 2008, the year is 2008 + 25=2033.

Answer:

a. 640.9
b. 2033