Question

At what rate is the diagonal of a cube increasing if its edges are increasing at a rate of 6.4 cm/s?

Answer 1

Answer:

11.08 cm/s

Step-by-step explanation:

Here it is given that the edges of a cube are increasing at a rate of 6.4 cm/s . And we need to find out the rate at which the diagonals are increasing . If we assume each side of the cube to be x , then ;

$\longrightarrow \dfrac{da}{dt}= 6.4 \ cm/s\dots (i)$

Now we can find diagonal as ,

$\longrightarrow d = \sqrt{ length^2+breadth^2+height^2}$

Here l = b = h = a ; so ,

$\longrightarrow d =\sqrt{ a^2+a^2+a^2}$

Simplify,

$\longrightarrow d =\sqrt{3a^2}$

Take out a from square root,

$\longrightarrow d =\sqrt3 a$

Now , we can find the rate of change of diagonal as ;

$\longrightarrow r =\dfrac{d\sqrt3 a}{dt}$

Take out the constant,

$\longrightarrow r = \sqrt3\dfrac{da}{dt}$

From equation (i),

$\longrightarrow r = 1.732 \times 6.4 \ cm/s$

Multiply ,

$\longrightarrow \underline{\underline{ rate_{of\ change \ of \ diagonals}= 11.08\ cm/s}}$

This is the required answer!

Answer 2

If the side length increases at a rate of 6.4 cm/s, then the diagonal increases at a rate of 11.1 cm/s

At what rate does the diagonal increase?

The diagonal of a cube of side length S is given by:

d = S*√3

In this case, we know that the side length increases at a rate of 6.4 cm/s, then we can write:

S = S₀ + 6.4cm/s*t

Where S₀ is the initial side length, replacing that on the diagonal formula we get:

d = (S₀ + 6.4cm/s*t)*√3

Now if we differentiate it with respect to the time t, we get:

d' = 6.4cm/s*√3 = 11.1 cm/s

So, if the side length increases at a rate of 6.4 cm/s, then the diagonal increases at a rate of 11.1 cm/s.

If you want to learn more about rates of change, you can read:

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