Advertisements
Advertisements
Question
If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?
Advertisements
Solution
Given, side is increasing at the rate of 4 cm/sec.
Let equal sides are of length x cm.
∴ `dx/dt` = 4 cm/sec
∵ Area of isosceles triangle is `b/2 sqrt(a^2 - b^2/4)`
∵ b = 10 cm and a = x
`\implies` A = `10/2 sqrt(x^2 - (10)^2/4`
`\implies` A = `5sqrt(x^2 - 25)`
`\implies (dA)/dt = 5/2 (2x)/sqrt(x^2 - 25) dx/dt`
`(dA)/dt = (5x)/sqrt(x^2 - 25) dx/dt`
∵ x = 10 ...(when all sides become equal)
`\implies (dA)/dt = (5 xx 10)/sqrt(100 - 25) xx 4`
= `(50 xx 4)/sqrt(75) = (50 xx 4)/(5sqrt(3))`
`(dA)/dt = 40/sqrt(3)`
APPEARS IN
RELATED QUESTIONS
The Volume of cube is increasing at the rate of 9 cm 3/s. How fast is its surfacee area increasing when the length of an edge is 10 cm?
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
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?
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) associated with the production of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?
A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.
A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?
Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep ?
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the perimeter.
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the area of the rectangle.
If a particle moves in a straight line such that the distance travelled in time t is given by s = t3 − 6t2+ 9t + 8. Find the initial velocity of the particle ?
The volume of a sphere is increasing at 3 cubic centimeter per second. Find the rate of increase of the radius, when the radius is 2 cms ?
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?
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
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 radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is
The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius = 7 cm and altitude 24 cm is
The volume of a sphere is increasing at 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is
If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to
In a sphere the rate of change of volume is
The rate of change of volume of a sphere with respect to its surface area, when the radius is 2 cm, is ______.
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate
The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is ______.
The rate of change of volume of a sphere is equal to the rate of change of the radius than its radius equal to ____________.
A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.
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.
