
Draw the graph of where . Use the graph to find when , and to find when . Then find .
Answer
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Hint: We see that the given equation of curve is in implicit form we write it in the explicit form of function . We assume 3 positive values and 3 negative values for . We find the corresponding functional values. We find the coordinates of points and join them to get the plot of the curve . We draw the lines and the ordinate of its point of intersection with the curve will be . We draw the lines and abscissa of its point of intersection with the curve will be .
Complete step-by-step answer:
We have the given equation of the curve in the question as
Here the condition is the non-negative constraints which means the curve is only in first quadrant. We see that it is in implicit form which means it is not expressed in the form . We write explicitly in the form as
We can see that above function is clearly not defined for We see that and are multiplied to each other and the product is 20. We can take the factors of 20 as assumed integral values of . We have the factors of 20 as 1,2,4,5,10,20. We take 5 values in equation and find the corresponding values of . We write the obtained values in the following table.
We plot the obtained points and join the points to obtain the plot of the curve with
We are also asked to use the graph to find when , and to find when and then find . So we draw the lines and and then observe only the first quadrant as there are only positive values and see where the lines intersect the curve.
We observe that the line intersects the curve at , so when we have . We also observe that the line intersects the curve at , so when we have . The required value from the question is .
Note: We can verify the points of intersection by putting them in the equation of the curve. The plot we obtained is the plot of a rectangular hyperbola whose standard equation is given by for some real constant . The rectangular hyperbola is special type of hyperbola obtained when we put the condition in the general equation of an hyperbola .
Complete step-by-step answer:
We have the given equation of the curve in the question as
Here the condition is the non-negative constraints
We can see that above function is clearly not defined for

We plot the obtained points

We are also asked to use the graph to find

We observe that the line
Note: We can verify the points of intersection by putting them in the equation of the curve. The plot we obtained is the plot of a rectangular hyperbola whose standard equation is given by
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