Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let ‘a’ be the side of an equilateral triangle.
Medain AM = `sqrt(3)/2a`
Given `(d(AM))/(dt) = 2sqrt(3)` cm/s
`(d(AM))/(dt) = (d(sqrt(3)/2a))/(da) xx (da)/dt`
`2sqrt(3) = sqrt(3)/2 . (da)/dt`
`\implies (da)/dt` = 4 cm/s
APPEARS IN
RELATED QUESTIONS
If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`
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 (a) the perimeter, and (b) the area of the rectangle.
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
A balloon, which always remains spherical, has a variable diameter `3/2 (2x + 1)` Find the rate of change of its volume with respect to x.
The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?
A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?
Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.
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 ?
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 ?
A particle moves along the curve y = (2/3)x3 + 1. Find the points on the curve at which the y-coordinate is changing twice as fast as the x-coordinate ?
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.
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
For what values of x is the rate of increase of x3 − 5x2 + 5x + 8 is twice the rate of increase of x ?
The coordinates of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are
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
In a sphere the rate of change of surface area is
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.
Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being seperated.
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
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
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 ____________.
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.
What does evaluating \[\left.\frac{dy}{dx}\right|_{x=x_0}\] give?
When \(x\) and \(y\) both vary with respect to \(t\), which formula gives the rate of change of \(y\) with respect to \(x\)?
The total cost in Rupees is \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\]. Which expression is the marginal cost?
