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Nuclear Plant Utilization The United States is not building many nuclear plants, but the ones that it has are running full tilt. The output (as a percent of total capacity) of nuclear plants is described by the equation $$ y=1.9467 t+70.082 $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1990 . a. Sketch the line with the given equation. b. What are the slope and the \(y\) -intercept of the line found in part (a)? c. Give an interpretation of the slope and the \(y\) -intercept of the line found in part (a). d. If the utilization of nuclear power continued to grow at the same rate and the total capacity of nuclear plants in the United States remained constant, by what year were the plants generating at maximum capacity?

Step by step solution

01

Sketch the line

To sketch the line represented by the given equation \[y = 1.9467t + 70.082\] we can use the slope-intercept form of the linear equation, which is \(y = mx + b\), where m is the slope and b is the y-intercept. The given equation is already in the slope-intercept form. Thus, we can plot the y-intercept on the y-axis (when t=0) and use the slope to find another point on the line. Then, draw a line connecting the two points.

02

Determine slope and y-intercept

The given equation has the form \[y = 1.9467t + 70.082\] In this equation, the slope, m, is 1.9467, and the y-intercept, b, is 70.082.

03

Interpret the slope and the y-intercept

The slope, m, represents the rate of change in the output percent of nuclear plants with respect to time. In this case, a slope of 1.9467 means that for every additional year, the output percent of nuclear plants increases by 1.9467% of total capacity. The y-intercept, b, represents the output percent when \(t = 0\), which corresponds to the beginning of 1990. Therefore, at the beginning of 1990, the output percent of nuclear plants was 70.082% of total capacity.

04

Estimate the year when utilization reaches maximum capacity

When the plants reach maximum capacity, their output will be 100%. We can set the equation to be equal to 100 and solve for t to determine when this will occur: \[100 = 1.9467t + 70.082\] Subtract 70.082 from both sides and divide by 1.9467: \[t = \frac{100 - 70.082}{1.9467} \approx 15.35\] This means that it will take approximately 15.35 years from the beginning of 1990. To find the year, we can add the elapsed time to 1990: \[1990 + 15.35 \approx 2005.35\] Thus, the plants were generating at maximum capacity around the year 2005.

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