Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
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 maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
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) = x3 − 3x
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) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
f(x) = ex
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find the maximum and minimum values of x + sin 2x on [0, 2π].
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?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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.`
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Divide the number 30 into two parts such that their product is maximum.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Solve the following : 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)`.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
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`
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
A function f(x) is maximum at x = a when f'(a) > 0.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, 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 ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
