Advertisements
Advertisements
प्रश्न
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
Advertisements
उत्तर
The height h at any t is given by h = 3 + 14t – 5t2
∴ `"dh"/dt = d/dt(3 + 14t - 5t^2)`
= 0 + 14 x 1 – 5 x 2t
= 14 – 10t
and `(d^2h)/(dt^2) = d/dt(14 - 10t)`
= 0 – 10 x 1
= – 10
The root of `"dh"/dt` = 0,
i.e. 14 – 10t = 0 is t = `(14)/(10) = (7)/(5)`
and
`((d^2h)/(dt^2))_("at" t = 7/5)` = −10 < 0
∴ By the second derivative test, h is maximum at t = `(7)/(5)`.
∴ Maximum height = `(3 + 14t – 5t^2)_("at" t = 7/5)`
= `3 + 14(7/5) - 5(7/5)^2`
= `3 + (98)/(5) - (245)/(25)`
= `(75 + 490 - 245)/(25)`
= `(320)/(25)`
= 12.8
Hence, the maximum height the ball can reach = 12.8 units.
APPEARS IN
संबंधित प्रश्न
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = sinx − cos x, 0 < x < 2π
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Prove that the following function do not have maxima or minima:
g(x) = logx
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
Range of projectile will be maximum when angle of projectile is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
