Use calculus to show that the maximum possible area of a triangle with one vertex at (0, 0) and the other two on the positive y-axis and the negative x-axis where the diagonal has length L = 3, occurs when the angles at both (other) vertices are π/4. Include a sketch.

Q: Show that the square of the length of subtangent at any point on the curve by2=(x+a)2 (b is not…

A: Here given curve  by2=x+a2

Q: Find the minimum distance from the point (5, 0, 0) to the paraboloid z = x2 + y2. (Give your answer…

A: Given:    z = x2 + y2  Point: ( 5, 0, 0 )

Q: 6. Find the shortest distance from B6,0,0) to the pane Ax + By tCz +D= 0 where A, B,C and D SR .

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Q: Find the minimum distance from the point to the plane x-y+z=3 at (1,-3,2)

A: We will find the minimum distance from the point to the plane x-y+z=3 at (1,-3,2) in the following…

Q: show that the square of the length of subtangent at any point on the curve by2=(x+a)3(b is not equal…

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Q: Using Lagrange multiplier, find the minimum value of f(x,y) = 3x +ly on the circle x2 + y2 = 1. Give…

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Q: Use Picard's theorem, find the largest interval for which the IVP y y(0) = 1, %3D %3D x² +x – 6' has…

A: Given that, y''=yx2+x-6,y(0)=1 Let F(x,y,y')=yx2+x-6 Then, ∂f∂y=1x2+x-6 Since,…

Q: Find the minimum distance from the point to the plane x - y + z = 3.

A: Now, formula of shortest distance of the plane ax+by+cz=d passing through the point (l, m, n) is,

Q: Find the points on the surfaces z = V² + y? and z = 2 – x/2 having the shortest distance to the…

A: We know that the two given surfaces will intersect at some curve whose equation can be obtained by…

Q: Find the minimum distance from the point (4, 0, 6) to the plane x − y + z = 3.

A: Find the minimum distance from the point (4, 0, 6) to the plane x − y + z = 3.

Q: Use Picard's theorem, find the largest interval for which the IVP y y(0) = 1, %3D x² +x – 6' has a…

A: Given that, y''=yx2+x-6,y(0)=1 Let F(x,y,y')=yx2+x-6 Then, ∂f∂y=1x2+x-6 Since,∂f∂y=0,…

Q: 2. A rectangle is bounded by the x-axis and the semicircle y = V16 – x². What length and width…

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Q: Find the arclength of y = 2x2/3 from x is greater than or equal to 1 and less than or equal to 2

A: The arc length of the function y=2x23 between x=1 to x=2.

Q: Find the length L and W (with W(less than or equal to) L) of the rectangle with perimeter 48 that…

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Q: (c) cut from each corner of length x cm, a maximum volume for the resulting open-top cake tin will…

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Q: In constructing an open rectangular box from 75 ft² of material, what dimensions will result in a…

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Q: Find the maximum value of f(x, y) = 4x + 6y subject to being on the circle = 13. 5. x² + y

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Q: Show that the square of the length of the subtangent at any point on the curve by2=(x+a)3 (b is not…

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Q: Find the minimum distance from the point to the plane x – y + z = 12. (Hint: To simplify the…

A: According to bartleby guidelines when student ask more than one question the expert should answer…

Q: Show that the minimum distance from the origin to a point on theplane ax + by + cz = d is

A: Given: ax+by+cz=d we know that minimum distance p of a point x1,y1,z1 from given equation of plane…

Q: What is the shortest distance from the point (4,0, 5) to the sphere (x – 1)2 + (y + 2)2 + z² = 36?…

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Q: Show that the sum of the squares of the distances from a pointP = (c, d) to n fixed points (a1, b1),…

A: Here, the objective is to show that the sum of the squares of the distances from a point P = (c, d)…

Q: A rectangle is bounded by the x-axis and the semi-circle y = √(36 − x^2). What length and width…

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Q: Find the minimum distance from the point to the plane x - y + z = 9. (Hint: To simplify the…

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Q: Find the minimum distance from the point to the plane x - y + z = 6. (Hint: To simplify the…

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Q: Find the minimum distance from the point (-6, -6, 0) to the surface z = O√√559 O 3√61 09 O√773 O√69…

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Q: Show that the square of the length of subtangent at any point on the curve by 2 = (x+a)3,(b is not…

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Q: Which of the following is the least distance from the point (2, –1,1) to the plane x + y – z = 2?

A: Given: The plane equation is x+y-z=2and the distance from the point is 2,-1,1 To find the least…

Q: Ue langrange multipliars to fined the Shortest distancé fam the to the pGine x+g+Z=1. point (7,0,)

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Q: In constructing an open rectangular box from 48 ftf of material, what dimensions will result in a…

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Q: A rectangle, R, has length L = 8 and width W = 3. What is the maximum area of the rectangle, Q, that…

A: We need to find maximum area.

Q: A tube of lip balm has a height of 2.75 inches and a diameter of 0.75inches.  a. if the cap on the…

A: Part (a):Diameter of the lip balm tube, d = 0.75 inches. Thus, the radius of the tube is r = d/2 =…

Q: A worker is to construct an open rectangular box with a square base and a volume of 147 ft3. If…

A: Given,Volume of the rectangular box = 147 ft3The cost of the bottom = $6 per ft2The cost of the…

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Q: Find the points on the cone 2 = r² + y? that are closest to the point (4, 2, 0). (Hint: You can…

A: Here we use the distance formula  D=(x2-x1)2+(y2-y1)2+(z2-z1)2 where D is the distance between the…

Q: Find the minimum distance from the point to the surface z = V7 - 2x – 2y. (Hint: To simplify the…

A: We need to find the minimum distance from the point A (-2,-2,0) to the surface z = 7-2x-2y…

Q: A worker is to construct an open rectangular box with a square base and a volume of 147 ft3. If…

A: Consider the given information about the box:

Q: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 1 –…

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Q: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 8 –…

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Q: Find the area of on of the triangle that can drawn between the planes Зх — бу- 2г 15 and 2x + y - 2z…

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Q: 3. A divided field is to be constructed with 4000 feet of fence as shown. For what value of z will…

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Q: A box with a square base and with height f(x) is to be constructed so that it has a volume of 1000…

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Q: Find the families of characteristics of the PDE (1-x²)w„ - u,, = 0 in the elliptic and hyperbolic…

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Q: Find the rectangle of the maximum area that can be inscribed in a right triangle with legs of length…

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Q: Find the maximum value of f(x, y) = 4xy(9 – 72x – 9y) over the triangle with vertices -- - (0,0), (5…

A: Given: fx,y=4xy9-72x-9y vertices 0,0 18,00,1

Q: What is the maximum vertical distance between the line y -x = 2 and the parabola y = x? in the…

A: We have to find the maximum vertical distance between the line y-x=2 and the parabola y=x2 in the…

Q: OFind the aquahon of the Hangent to the at point Lircle X2 +8Xt PCI,3)

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

Q: A worker is to construct an open rectangular box with a square base and a volume of 48 ft3. If…

A: Note: Please check the material cost for side. Because the dimension seems to be unrealistic since…