Zachary Peace
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Few Questions (1 Viewer)
- Thread starter Zachary Peace
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Zachary Peace
New Member
- Joined
- Aug 12, 2017
- Messages
- 5
- Gender
- Male
- HSC
- 2018
One more question,
At a particular location, a river 30 metres wide is measured for depth every 5 metres across its width. The measurements from bank to bank are given in the following table
(i) Use simpsons rule to find the cross-sectional area of the river at this point.
(ii) Use your answer from (i) to find the volume of the water passing through this point in 2 hours, if the water is passing this spot at 1/8 m/sec
At a particular location, a river 30 metres wide is measured for depth every 5 metres across its width. The measurements from bank to bank are given in the following table
(i) Use simpsons rule to find the cross-sectional area of the river at this point.
(ii) Use your answer from (i) to find the volume of the water passing through this point in 2 hours, if the water is passing this spot at 1/8 m/sec
1)
Straight away this integral must be negative, so the answer is either A or D.
)
appears to be symmetric around the y-axis, so
\, dx = 2 \int _{0}^2 f(x)\, dx )
You are given
+g(x)\, dx = \int _0^2 f(x)\, dx+\int _0^2g(x)\, dx = \pi - 10)
and note that
\, dx )
represents the area of the semicircle from 0 to 2, which you can find in exact form.
2))
A quadratic is positive definite (i.e. always greater than zero) if:
This restriction on the leading coefficient (
) eliminates two of the possible answers immediately.
You can do a simple substitution test on the remaining two choices, which are
and 
.
If the quadratic is positive for all values of
, it must be positive for 
. Substituting , we are left with just the constant term 
, which has to be positive.
3)
i) If you haven't already, use Simpson's rule. I like to remember it as
\, dx \approx \frac{h}{3}[2(y_1+y_3+y_5 +...) + 4(y_2 +y_4+y_6 +...)] )
Where
is the interval length.
ii) A diagram really helps to understand the problem.
Every second, 1.8m of water passes through the cross section.
You can imagine a 3D prism 1.8m long, whose cross section is that of the river's. That's how much water passes through every second.
So, the volume of water passing through the cross section each second is given by
\left( \int f(x) \, dx\,\mathrm{m^2}\right) = \left( 1.8 \int f(x) \, dx\,\right) \mathrm{m^3/s})
And to get the amount in 2 hours, just do another multiplication.
Straight away this integral must be negative, so the answer is either A or D.
You are given
and note that
represents the area of the semicircle from 0 to 2, which you can find in exact form.
2)
A quadratic is positive definite (i.e. always greater than zero) if:
- it has no real roots, and
- its leading coefficient is positive.
This restriction on the leading coefficient (
You can do a simple substitution test on the remaining two choices, which are
If the quadratic is positive for all values of
3)
i) If you haven't already, use Simpson's rule. I like to remember it as
Where
ii) A diagram really helps to understand the problem.
Every second, 1.8m of water passes through the cross section.
You can imagine a 3D prism 1.8m long, whose cross section is that of the river's. That's how much water passes through every second.
So, the volume of water passing through the cross section each second is given by
And to get the amount in 2 hours, just do another multiplication.
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