Concept explainers
Projectile Motion In Exercises 81 and 82, consider a projectile launched at a height h feet above the ground and at an angle _ with the horizontal. When the initial velocity is v0 feet per second, the path of the projectile is modeled by the parametric equations
A rectangular equation for the path of a projectile is
(a) Eliminate the parameter I from the position function for the motion of a projectile to show that the rectangular equation is
(b) Use the result of part (a) to find h,
(c) Use a graphing utility to graph the rectangular equation for the path of the projectile. Confirm your answer in part (b) by sketching the curve represented by the parametric equations.
(d) Use a graphing utility to approximate the maximum height of the projectile and its range.
Want to see the full answer?
Check out a sample textbook solutionChapter 10 Solutions
Calculus
- Use a graphing utility to graph the curve represented by the parametric equationsHypocycloid: x = 3 cos3 θ, y = 3 sin3 θ . Indicate the orientation of the curve. Identify any points at which the curve is not smooth.arrow_forwardUse a graphing utility to graph the curve represented by the parametric equations. (Indicate the orientation of the curve.) x = 6 sin(2 theta), y = 4 cos(2 theta)arrow_forwardDiscuss the difference between parametric and non-parametric tests. Justify your answer with example.arrow_forward
- The center field fence in a ballpark is 7 feet high and 408 feet from home plate. A baseball is hit at a point h = 2.5 feet above the ground. It leaves the bat at an angle of ? degrees with the horizontal at a speed of 110 miles per hour. Write a set of parametric equations that model the path of the baseball. (Let t be in seconds and let x and y be in feet.) Find the minimum angle required for the hit to be a home run. (Round your answer to one decimal place.)arrow_forwardFind the parametric equations of the intersection of the planes x + (y − 9) + z = 0 and −x + (y + 9) − z = 0. Use the parameter t. Enter your answers as a comma-separated list of equations.)arrow_forwardYou are flying in a small airplane at an altitude of 6161 feet. From that second, your horizontal air speed is 253 feet per second and your rate of ascent is 31 feet per second.Write a set of parametric equations for the plane's ascent where t=0 seconds is when the plane is at an altitude of 6161 feet. x(t)=_____ y(t)=_______ How far will you have traveled horizontally during the time you ascend from 6161 feet to 6626 feet? _____________feet.arrow_forward
- True or False? In Exercises 115-117, determine whether the statement is true or false. Justify your answer. Only one set of parametric equations can represent the line y=32x.arrow_forwardFind a set of parametric equations for the tangent line to the curve of intersection of the surfaces at the given point. (Enter your answers as a comma-separated list of equations.) x2 + z2 = 25, y2 + z2 = 25, (3, 3, 4)arrow_forwardConsider the following set of parametric equations. Clearly sketch the plane curve that they describe using our ‘Analysis of Intervals’ technique. Label in your graph all intercepts and points at which the tangent is vertical or horizontal. In addition, write, but do not evaluate, two integrals. One which would determine the area between the two x- intercepts and one which would determine the length of the plane curve that lies between the two x-intercepts.arrow_forward
- Find a set of parametric equations for the line passing through the point (2, 0, 5) that is parallel to the plane given by x + y + z = 5, and perpendicular to the line x = −1 + t, y = 1 + t, z = 6 − t. (Enter your answers as a comma-separated list.)arrow_forwardUse a graphing utility to graph the curve represented by the parametric equations. (Indicate the orientation of the curve.)arrow_forwardparametric equationsarrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage