Jim is running along a parabolic track where his position can be described as y = 3x2. He knows that his friend is waiting for him at position (0, 1), which is not on the track. Where will Jim be the closest to his friend?

Q: Determine the general equation of a parabola opening downwards whose vertex is at (3,-4) with a…

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Q: At what point does the line normal to the graph of y=5−4x+4x2 at (1,5) intersect the parabola a…

A: The given function is- y=5-4x+4x2.................1 Differentiating the above expression w.r.t. x-…

Q: At what point does the normal to y = 2 + 2x + 2x² at (1,6) intersect the parabola a second time?

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Q: Find the length of the curve 36y2 = (x — 4)3 , 2 ≤ y ≤ 3, y ≥ 0.

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Q: At what point does the normal to y=4x2−(2+3x)y=4x2−(2+3x) at (1,−1)(1,−1) intersect the parabola a…

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Q: Find the equation in the xy-plane whose graph includes x In(6t) and y = t².

A: Given  x= ln(6t) y=t2

Q: Find the standard equation of the parabola with x intercepts at x = 2 and x -3, and a y-intercept at…

A: Solution is given below

Q: Find a parabola with equation  y = ax2 + bx + c  that has slope  13  at x = 1, slope  −19  at x =…

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Q: Find the points on the ellipse 2x + 4y = 1 where f(x.y) = xy has its extreme values.

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Q: A particle follows the trajectory x(t) = t3 + 2t, y(t) = 20t – t2 with t measured in seconds and…

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Q: The surface of a mountain is modeled by the equation h(x, y) = 5000 − 0.001x2 − 0.004y2. A mountain…

A: Hints:  The given equation is h(x,y)=5000-0.001x2-0.004y2 and the given point is (500,500,3750). To…

Q: What is the general equation of a parabola that opens upward with V(-1,2)?

A: The general equation of the parabola opening upwards is x-h2=4py-k where (h,k) is vertex. here…

Q: Two children weighing 18 and 20 kilograms are sitting on opposite sides of a seesaw, both 2 meters…

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Q: Find all points on the curve 5x² + 8y² = 64 %3D

A: Given: The curve 5x2+8y2=64 where tangent line passes through the point (11, 0). To find: All the…

Q: In the xy-plane, a line that has the equation y = c for some constant c intersects a parabola at…

A: in the xy-plane a line that has the equation y=c for some constant c intersects a parabola at…

Q: Find the equation of parabola “y = ax2 + bx + c” passing through points (1, -6), (2, -3), (-1, 0)

A: Consider the given points and the equation of the parabola. Obtain the system of equations with a, b…

Q: What is the measure of the focal width of the parabola y+2y+4x-3=0 ?

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Q: The surface of a mountain is modeled by the equation h(x, y) = 7000 - 0.001x2 - 0.004y2. A mountain…

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Q: If a person shoots a basketball overhand from a position 8 feet above the​ floor, then the path of…

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Q: Find the point on the parabola y* = 2x that is nearest the point (1, 4).

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Q: . What is the snortest distance between the parabola y=2x² and the line y=4x-4

A: Shortest distance between two curves are always along the common normal.

Q: 2. What is the value of y anytime a parabola intersects the x-axis?

A: It is known that the value of y on x-axis is always 0.

Q: Find the length of the curve x = 1+ 312, y = 4 + 213, 0 <t < 1. Length =

A: Given,

Q: -16х2 %3D 0.117 р2 y +2.75х + 3

A: Given, v =  29.97285 y = -16x20.117v2 + 2.75x +3 y at  v =  29.97285 y = -16x20.117(29.97285)2 +…

Q: Find the equation of the tangent to the hyperbola x2 - 4y2 = 9 at (5, 2).

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Q: Find the points on the ellipse  X^2 + 6y^2 + 3xy =40 With the greatest x coordinator

A: Let us assume thatfx,y=xandgx,y=x2+6y2+3xy-40The lagrange multiplier equations…

Q: A baseball is launched into the outfield on a parabolictrajectory given by y = 0.01x(200 - x). Find…

A: Given ------(1)   The average value of a functionon an interval is given by ------ (2) From the…

Q: Find the equation for the parabola that has its vertex at the 1 origin and has directrix at y 41…

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Q: Find the standard equation of the parabola with directrix x + 1 = 0, axis y = 1 and the length of…

A: Find the standard equation of the parabola with directrix x + 1 = 0, axis y = 1 and the length of…

Q: Find equations for two lines through the origin that are tangent to the ellipse a- 8x +y + 3 =0.

A: We are asked to find the equation of two lines that are tangent to the ellipse.

Q: Find the coordinates of the point on the parabola y = x² which is closest to the point (3, 0).

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Q: Sketch the graph of the cylinder given by 4x2 + 9y2 = 36. show solution

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Q: Find equations for two lines through the origin that are tangent to the ellipse x2 – 8x + y² + 3 =…

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Q: Find the points on the ellipse 2x +y = 1 where f(x,y) = xy has its extreme values.

A: Introduction: When a multivariable function is to be extremized under some constraints, we use the…

Q: FInd the equations of the tangents with slope m=-2/9 to the ellipse 4x^2+9y^2=40

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Q: The equation of the line touching both the parabolas y = 4x and x' =-32y is x-2y+1 = 0, then a is

A: Introduction :

Q: What is the minimum vertical distance between the parabola y =x2 +1and y =x -x2 ?

A: Given:

Q: Find equations for two lines through the origin that are tangent to the ellipse x - 8x + y² + 5 = 0.

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Q: Find the points of intersection of the parabola with equation 2x + y² + 4y – 2 = 0 and the line…

A: Here we find the intersecting point of parabola and the line.

Q: Find the smallest outputs of the function x^2+2y^2 on the slanted ellipse x^2+y^2+xy=3

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Q: What are the coordinates of the center of the curve x2 + y2 – 2x – 4y – 31 = 0?

A: Given, To find coordinates of center for the curve x2 + y2 – 2x – 4y – 31 = 0?

Q: Find the coordinates of the point in the curve y = 12x – 2y + 23 that is nearest to the point P(10,…

A: Given curve: y2=12x-2y+23 The point is  P10,4 To find: The coordinates of the point on the curve…

Q: At what point does the normal to y = 2x2 – (3 + 2x) at (1, –3) intersect the parabola a second time?…

A: Given: y=2x2-3+2x Point: 1,-3 Objective is to find the point where the normal to y=2x2-3+2x at…

Q: A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating…

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Q: Jim is running along a parabolic track where his position can be described as y = He knows that his…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the equation of the parabola with directrix y = 3, axis x = 0, and latus rectum 4

A: Given: Equation of directrix:y=3……1Equation of axis:x=0……2Length of latus rectum:4……3

Q: Find the equation of the parabola with directrix x + 2 = 0, axis y = 4, and latus rectum 4.

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Q: Find the point P on the parabola y = x2 closest to the point (57, 0). (х, у) %3D ]-«

A: Given the equation of parabolay=x2         ........(1)

Q: What is the maximum height of a particle whose path has parametric equations x = t9, y = 4 − t2?

A: The height of the particle is given by it's y position: y = 4 - t2 At the maximum height, dy/dt = 0

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.