Volume of a Fish Tank A fish tank in an avant-garde...

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is 300 $\mathrm{cm}$ long and 120 $\mathrm{cm}$ wide. The front and back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure 120 $\mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.) (a) Let $a, b,$ and $c$ be the three vectors shown in the figure. Find $a \cdot(b \times c) .[\text { Hint: Recall that } \mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$ and $|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta . ]$ (b) What is the capacity of the tank in liters? [Note: $1 \mathrm{L}=1000 \mathrm{cm}^{3} .$]

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is 300 $\mathrm{cm}$ long and 120 $\mathrm{cm}$ wide. The front and back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure 120 $\mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.)
(a) Let $a, b,$ and $c$ be the three vectors shown in the figure. Find $a \cdot(b \times c) .[\text { Hint: Recall that } \mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$ and $|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta . ]$
(b) What is the capacity of the tank in liters? [Note: $1 \mathrm{L}=1000 \mathrm{cm}^{3} .$]

Key Concepts

Volume Calculation of a Parallelepiped

Calculating the volume of a parallelepiped involves finding the absolute value of the triple scalar product of vectors along its edges. This approach is essential in various applications, linking vector operations with geometric volume determination.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar, calculated as the product of the two vectors' magnitudes and the cosine of the angle between them. It measures the extent to which two vectors are parallel and is fundamental in projecting one vector onto another.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector perpendicular to the plane containing the original vectors. The magnitude of the cross product is the product of the magnitudes of the original vectors and the sine of the angle between them, providing a measure of area spanned by the vectors.

Triple Scalar Product

The triple scalar product combines both the dot and cross products, defined as a dot product of one vector with the cross product of the other two. It computes a scalar and, in geometric contexts, its absolute value gives the volume of a parallelepiped formed by the three vectors.

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units into an equivalent value in another unit system. In problems involving volumes, this often involves converting from cubic centimeters to liters, using the relationship that 1000 cubic centimeters equal one liter.

Transcript

00:01 This question describes a fish tank with a funky shape and asks us to find first the scalar triple product of the three edges and then the volume in liters.

00:13 So let's go ahead and start.

00:17 The dimensions that we're given, we are told that the bottom, which is a rectangle, has a width of 120.

00:24 That's described by vector a.

00:26 So the length of vector a is 120 and a length of 300.

00:33 That's vector c.

00:34 So we're going to skip b for now.

00:36 Vector c is 300 centimeters.

00:41 Then it has the edges on the front and back are straight up and down.

00:47 So that's a 90 degree angle.

00:48 But on the side are slanted in at 30 degrees from the vertical.

00:54 So b, which is the length of the slanted side, is the magnitude of b, we're told, is 150 centimeters.

01:08 So it's a funky shape, and i'm going to sort of try to sketch it out.

01:13 We have a and c here, and it's a right angle.

01:18 So i imagine that's a rectangle.

01:21 And then we have b coming up like this here.

01:25 And it's it would almost look like that if it were sort of okay we'll pretend that's a good drawing so what we're asked to do first is find the scalar triple product of a b and c we're asked to find a dot b cross c now we know we don't have any of the actual vectors but we do have formulas for the magnitude we we know that a .b cross c is equal to the magnitude of a times the magnitude of b cross c times the cosine of the angle in between.

02:10 And i'm going to call the angle in between.

02:14 I'm just going to call that theta 1 because we'll have to deal with other angles further on, but we'll call this one theta 1.

02:22 Now we know a already, 120 centimeters.

02:26 B, we have a formula 4.

02:28 We know that b cross c has a magnitude of b, the magnitude of b, times the magnitude of c, times the sign of the angle in between.

02:41 And now this is a different angle than theta 1.

02:43 I'll call this angle theta 2.

02:45 And we'll come back to that.

02:47 But first we want to figure out what a is here.

02:50 So we have a and the magnitude of b cross c.

02:53 We also have cosine of theta 1.

02:54 Now theta 1 is the angle between a and b cross c.

02:59 It's kind of hard to tell on our diagram, but b cross c, if this is the side face, b cross c would be coming straight out of the screen.

03:14 So it's b cross c, the side face is what the screen is parallel to the screen.

03:21 If that's b and c, b cross c would be coming straight out or going straight in.

03:25 We also are told that since the bottom is a rectangle, a would be going straight in or straight out of the page.

03:33 So b cross c and a are actually parallel, which means the angle in between them is zero...

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