a)Given \(\displaystyle{x}={20}{\cos{{t}}},{y}={10}{\sin{{t}}}\)

To find the cartesian equation that represents the race track the car is going on solve for t using the equation of y.

\(\displaystyle{y}={10}{\sin{{t}}}\)

\(\displaystyle\frac{{y}}{{10}}={\sin{{t}}}\)

\(\displaystyle{t}={{\sin}^{{-{1}}}{\left(\frac{{y}}{{10}}\right)}}\)

Further substitute the value of t in the equation of x.

\(\displaystyle{x}={20}{\cos{{\left({{\sin}^{{-{1}}}{\left(\frac{{y}}{{10}}\right)}}\right)}}}\)

Thus we have found the cartesian equation that represents the race track the car is going on.

b)The parametric equations we would use to make the car go three times as faster on the same track is

\(\displaystyle{x}{\left({t}\right)}={20} {\cos{{t}}}{\left({3}{t}\right)}, {y}{\left({t}\right)}={10} {\sin{{\left({3}{t}\right)}}}\)

c)The parametric equations we would use to make the car go half as fast on the same track is \(\displaystyle{x}={20}{\cos{{\left(\frac{{t}}{{2}}\right)}}},{y}={10}{\sin{{\left(\frac{{t}}{{2}}\right)}}}\)

To find the cartesian equation that represents the race track the car is going on solve for t using the equation of y.

\(\displaystyle{y}={10}{\sin{{t}}}\)

\(\displaystyle\frac{{y}}{{10}}={\sin{{t}}}\)

\(\displaystyle{t}={{\sin}^{{-{1}}}{\left(\frac{{y}}{{10}}\right)}}\)

Further substitute the value of t in the equation of x.

\(\displaystyle{x}={20}{\cos{{\left({{\sin}^{{-{1}}}{\left(\frac{{y}}{{10}}\right)}}\right)}}}\)

Thus we have found the cartesian equation that represents the race track the car is going on.

b)The parametric equations we would use to make the car go three times as faster on the same track is

\(\displaystyle{x}{\left({t}\right)}={20} {\cos{{t}}}{\left({3}{t}\right)}, {y}{\left({t}\right)}={10} {\sin{{\left({3}{t}\right)}}}\)

c)The parametric equations we would use to make the car go half as fast on the same track is \(\displaystyle{x}={20}{\cos{{\left(\frac{{t}}{{2}}\right)}}},{y}={10}{\sin{{\left(\frac{{t}}{{2}}\right)}}}\)