Advertisements
Advertisements
प्रश्न
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. How far is the area increasing when the side is 10 cms?
Advertisements
उत्तर
\[\text { Let x be the side andAbe the area of the equilateral triangle at any timet.Then, }\]
\[A = \frac{\sqrt{3}}{4} x^2 \]
\[ \Rightarrow \frac{dA}{dt} = 2 \times \frac{\sqrt{3}}{4} x^{} \frac{dx}{dt}\]
\[\Rightarrow\frac{dA}{dt}=\frac{\sqrt{3}}{2}\times2\times10\]
\[\Rightarrow\frac{dA}{dt}=10\sqrt{3} \text {cm}^2 /sec\]
APPEARS IN
संबंधित प्रश्न
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
The total cost C(x) in rupees associated with the production of x units of an item is given by C(x) = 0.007x3 – 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is ______.
Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies?
Find the rate of change of the volume of a sphere with respect to its diameter ?
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?
Find the rate of change of the volume of a cone with respect to the radius of its base ?
Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?
The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.
If y = 7x − x3 and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?
A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9 cm.
A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?
The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.
The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate of change of its surface area at the instant when radius is 5 cm ?
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?
A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall ?
Side of an equilateral triangle expands at the rate of 2 cm/sec. The rate of increase of its area when each side is 10 cm is
The volume of a sphere is increasing at the rate of 4π cm3/sec. The rate of increase of the radius when the volume is 288 π cm3, is
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is
If s = t3 − 4t2 + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, is
In a sphere the rate of change of volume is
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
The radius of a cylinder is increasing at the rate of 3 m/s and its height is decreasing at the rate of 4 m/s. The rate of change of volume when the radius is 4 m and height is 6 m, is ____________.
A particle is moving along the curve x = at2 + bt + c. If ac = b2, then particle would be moving with uniform ____________.
The rate of change of area of a circle with respect to its radius r at r = 6 cm is ____________.
Total revenue in rupees received from the sale of x units of a product is given by R(x)= 3x2+ 36x + 5. The marginal revenue, when x = 15 is ____________.
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
A particle moves along the curve 3y = ax3 + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.
If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.
The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.
For the Chain Rule for Rates \[\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\], which condition is required?
For \[A=\pi r^2\], what is the rate of change of area with respect to radius when \[r=5\] cm?
For \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\], what is the marginal cost when \[x=3\]?
