Question

The roof of a rectangular floor (16m x 12m) base structure can be modelled by

z(x, y)=4+\frac{y}{25}-\frac{x^{2}}{50}

(x, y) is the coordinate of the floor with (0,0) as the coordinate of the centre of the floor, i.e. X = -8 to 8 and y = -6 to 6.

Use the online mathematical tools Wolfram Alpha to compute below (a), (b), (d) and (e).

Use Matlab (integral2 function) to compute below (f) and (g).

Show and explain the working steps, particularly on how those upper and lower limits are obtained for (d) and (e), together with the screen captures in your solutions.

Use Wolfram Alpha to calculate the volume of the structure by using both double integrals.

Use Wolfram Alpha to show the 3D graph of the roof (i.e.function z) above the floor region -8 < x < 8 and-6 < y<6.

\int_{a}^{b}\left[\int_{c}^{d} z(x, y) d x\right] d y=\int_{a}^{b} \int_{c}^{d} z(x, y) d x d y

\int_{e}^{f}\left[\int_{g}^{h} z(x, y) d y\right] d x=\int_{e}^{f} \int_{g}^{h} z(x, y) d y d x

Not using any mathematical tools, manually compute the volume of the structure by using below double integrals.

\int_{a}^{b}\left[\int_{c}^{d} z(x, y) d x\right] d y=\int_{a}^{b} \int_{c}^{d} z(x, y) d x d y

There is a pillar as shown below at the centre of the structure from the floor to the roof. Find the volume of the pillar by using below double integrals via Wolfram Alpha.Show and explain the working steps, particularly on how those upper and lower limits are obtained.

\int_{a}^{b} \int_{h(y)}^{H(y)} z(x, y) d x d y

By using the reversed order double integrals to find the volume of the pillar in (d) via Wolfram Alpha. Show and explain the working steps, particularly on how those upper and lower limits are obtained.

\int_{c}^{d} \int_{g(x)}^{G(x)} z(x, y) d y d x

Using Matlab (integral2 function) to find the volume of the pillar in (d) by using below double integrals. Use those limits that you have found in (d).

\int_{a}^{b} \int_{h(y)}^{H(y)} z(x, y) d x d y

g)pillar in (d) by using the reversed order double integrals.Use those limits that you have found in (e).Using Matlab (integral2 function) to find the volume of the

\int_{c}^{d} \int_{g(x)}^{G(x)} z(x, y) d y d x

By using the basic geometric method to estimate the volume of the pillar.

Verified

### Question 44046

During the summer months, British Columbia experiences numerous forest fires. You are working as a fire ranger in the interior of British Columbia and you spot a fire that is 15 km away from your tower. Your friend is working at another tower that is located 20 km from you at an angle of 21°. She can also see the fire.
b) Your friend decides to dispatch a water plane to the fire. Your station is not equipped with waterplanes. At what angle should the plane fly to target the fire accurately? (AP3)
Your friend decides to dispatch a water plane to the fire. Your station is not equipped with water planes. At what angle should the plane fly to target the fire accurately? (AP3)

### Question 44045

As a computer engineer, you are always looking at ways to improve the computing experience. As part of your final paper to submit for your Master's Degree, you do some research about how far computing has come over the years.
You make some interesting discoveries. Dating back to 1940, you find out that one of the original large computers was able to perform about 100 operations per second. After examining the data,you see that the speed of computers has multiplied 5-fold about every 7 years.
a) Some of the data is missing. You know that some super computers can now process4 882 812 500 operations per second. How many years have gone by to reach this number of operations? What year would this have occurred in? Show your solution algebraically! (AP5)
b) Do you think this is a realistic model? What limitation(s) may arise with this model? (CM3)

### Question 43374

Find the distance between the parallel planes P: 5x - 3y + z=1 and P: 10x-6y +2z=4. Enter an exact answer, do not use decimal approximation.

### Question 43373

Find an equation for the plane through the point (3, 2,-5) and parallel to the plane defined by5x- 4y + 4z = 12.
1=5(x-3)-4(y+5)+10(z-2)
0=-4(x+3)+5(y-2)+4(z+5)
1=5(x-3)-4(y-2)+4(z+5)
0=5(x-3)-4(y-2)+4(z+5)

### Question 43372

\text { Find the plane containing the three points } P(3,-2,1), Q(1,-3,0) \text { and } R(3,-5,5) \text {. }
0=-7(x+3)+8(y-2)+6(z+1)
0=-7(x-3)+8(y+2)+6(z-1)
0=-7(x-3)-8(y+2)+6(z-1)
1=-7(x-3)+8(y+2)+6(z-1)

### Question 43371

Find the directions of maximum and minimum change of f at the given point, and the values of the maximum and minimum rates of change.
f(x, y)=5 y^{2} e^{7 x},(4,-3)
\text { The maximum change is } 15 e^{28} \sqrt{445} ; \text { in the direction }\left(-315 e^{28}, 30 e^{28}\right)
\text { The minimum change is }-15 e^{28} \sqrt{445} ; \text { in the direction }\left(315 e^{28},-30 e^{28}\right) \text {. }
\text { The maximum change is } 15 e^{-28} \sqrt{445} ; \text { in the direction }\left(-315 e^{-28}, 30 e^{-28}\right)
\text { The minimum change is }-15 e^{-28} \sqrt{445} ; \text { in the direction }\left(315 e^{-28},-30 e^{-28}\right)
\text { The maximum change is } 15 e^{28} \sqrt{445} ; \text { in the direction }\left(315 e^{28},-30 e^{28}\right\rangle
\text { The minimum change is }-15 e^{28} \sqrt{445} ; \text { in the direction }\left\{-315 e^{28}, 30 e^{28}\right)
\text { The maximum change is } 15 e^{-28} \sqrt{445} ; \text { in the direction }\left(315 e^{-28},-30 e^{-28}\right)
\text { The minimum change is }-15 e^{-28} \sqrt{445} \text {; in the direction }\left(-315 e^{-28}, 30 e^{-28}\right\rangle

### Question 43370

Select the description and the sketch of the domain of the function.
f(x, y)=\frac{4 x y}{\left(y-x^{2}\right)}

### Question 43369

\text { Find an equation of the line passing through the points } P(3,4,-2) \text { and } Q(5,-4,3) \text {. }
\frac{x+3}{2}=\frac{y+4}{-8}=\frac{z-2}{5}
\frac{x+4}{-4}=\frac{y+4}{5}=\frac{z-3}{3}
\frac{x-3}{2}=\frac{y-4}{-8}=\frac{z+2}{5}
x-3=y-4=z+2

### Question 43368

Round each component to four decimal places, if necessary.
\text { Find the gradient of } f(x, y)=3 e^{5 x d y}+2 x \text { at }(-1,-1)
\nabla f(-1,-1) \approx

### Question 43367

\text { Find the gradient of the function } f(x, y)=x^{2}+4 x y^{2}-y^{7}