Rate of change question (1 Viewer)

[Ryuu-jin-jakka]

i) An equilateral triangle has sides of length s cm. Find an expression for the area, A cm^2, in terms of s

ii) The area of the triangle in part (i) is increasing at the rate of 4cm^2 per second. Find the exact rate, ds/dt, at which each side of the triangle is expanding when it has a side length of 2cm

 

[Timothy.Siu]

i) An equilateral triangle has sides of length s cm. Find an expression for the area, A cm^2, in terms of s

ii) The area of the triangle in part (i) is increasing at the rate of 4cm^2 per second. Find the exact rate, ds/dt, at which each side of the triangle is expanding when it has a side length of 2cm

i) A=1/2 x sin (pi/3) x s^2=root3/4 x s^2

ii) dA/dt=4
ds/dt=dA/dt x ds/dA

from i) dA/ds = root3/2 x s
ds/dA=2/(root3.s)

therefore ds/dt=4 x 2/(root3.s)
when s=2, ds/dt=4/root3=4root3/3 (cm/s)

 

[Ryuu-jin-jakka]

nice...

 

[Ryuu-jin-jakka]

3. Triangle ABC is isos., with AB = CB = 6cm and <ABC = #(theta)

i) Just proof that A = 18sin#(theta) - already done that
ii) Show the area of triangle ABC is incrasing at the raete of 3cm^2 per second, find the rate at which # is chaniging at the instant when the area of triangle ABC is 9cm^2

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5. A sector of a circle with centre O and radius r cm is bounded by radii OA and OB, and the arc AB. The angle substended at the centre of the sector is #, measured in radians
i) If r and # vary in such a way that the area of sector AOB has a constant value of 50cm^2, show that # = 100/r^2

ii) The radius of the sector is increasing at a rate of 0.5 cm per second. Find the rate at which the sector angle # is decreasing when the radius is 20 centimetres

 

[Ryuu-jin-jakka]

# = theta

 

[ianc]

3. Triangle ABC is isos., with AB = CB = 6cm and <abc>
i) Just proof that A = 18sin#(theta) - already done that
ii) Show the area of triangle ABC is incrasing at the raete of 3cm^2 per second, find the rate at which # is chaniging at the instant when the area of triangle ABC is 9cm^2

okay so the question gives us




from the first part, we have



differentiating:



and so just rearranging:



using part i, work out what # is when the area is 9cm^2, then substitute in to d#/dt


</abc>

<abc> 5. A sector of a circle with centre O and radius r cm is bounded by radii OA and OB, and the arc AB. The angle substended at the centre of the sector is #, measured in radians
i) If r and # vary in such a way that the area of sector AOB has a constant value of 50cm^2, show that # = 100/r^2

ii) The radius of the sector is increasing at a rate of 0.5 cm per second. Find the rate at which the sector angle # is decreasing when the radius is 20 centimetres

this question is much the same

for part i, use the formula for area of a sector



then part ii gives you dr/dt

so work out d#/dr (from part i), and combine these to work out d#/dt


hope this helps
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