What's the total price of TI-84 calculators sold in January?
241. A triangle has a 13-inch side, a 14-inch side, and a 15-inch side. To the nearest tenth of an inch, how long is the median drawn to the 14-inch side?
Q1. Which rule is helpful in triangles where we know the sidelengths?
Q2. What is the relationship between $\cos(\theta)$ and $\cos(180\deg-\theta)$?
Apply the Law of ... to two triangles in your diagram.
The triangle below has two fixed sides, joined by a "hinge" at point $A.$ Toggle the position of point $B$ to see how changing the third sidelength affects the median.
What is the length of the median drawn to side $c$ in a triangle with sidelengths $a,b$ and $c$?
242. In January and February, Edith’s Calculator Shop recorded the sales data shown below in the right-hand $3\times 2$ matrix S. The prices for these models are shown below in the left-hand $1\times 3$ matrix P. What is the meaning of the entries in the matrix product PS? Does the matrix product SP make sense?
Q1. Can any two matrices be multiplied?
Q2. What's the dimension of the matrix product PS?
Here's a visual explanation of matrix multiplication by 3Blue1Brown – you will never forget how to multiply matrices after you see this! You can also learn and practice matrix multiplication on Brilliant.
243. (Continuation) Edith decides to lower all calculator prices by 10%. Show how this can be done by multiplying one of the matrices above by a suitable scalar.
Q1. Which matrix must be modified to account for the price change?
244. A cylinder rests on top of a table, with a cone inscribed within, vertex up. Both heights and radii are 8 cm. A hemispherical bowl of radius 8 cm rests nearby on the same table, its circular rim parallel to the table.
Consider that part of the cylinder that is outside (above) the cone. Slice this region by a plane that is parallel to the table and 3 cm from it. The intersection is a ring between two concentric circles. Calculate its area.
The same plane slices the hemisphere, creating a disk. Show that the disk has the same area as the ring. The diagram shows both a top view and a side view of the ring, the disk, and the hemisphere. Note that point B is on the table.
Q1. One point is equidistant from all points of the hemisphere: where is it?
Express the area of the ring as a difference of two areas you can find.
On the hemisphere's side, find a right triangle with two known sidelengths.
245. Let $r$ be the common radius of the cone, cylinder, and hemisphere. Show that the ring and the disk have the same area, for all positions of the slicing plane.
Redo the previous problem replacing 3 and 8 with $h$ and $r$, respectively.
On the cone's side, consider a segment tangent to the inner circle with ends on the outer circle. Show that this segment is equal to the diameter of the disk (on the hemisphere's side.)
Furthermore, the disk and the ring have equal areas as you've shown in the problem. This illustrates a remarkable property of the ring (usually called an
Slide points $C_1$ and $C_2$ in the diagram below and compare the areas of the two rings:
Prove that the area of an annulus depends only on the length of its tangent chord. Find the length of the red chord in the diagram.
👀 Reveal answer:
$\frac{12}{\sqrt{\pi}}$
A beautiful consequence: Find the area bounded by the incircle and circumcircle of any regular polygon with sides of length 1.
🫣 Reveal answer:
246. (Continuation) If the hemispherical bowl were filled with liquid, it could be poured into the cylinder, which still has the cone inscribed in it. Will all the liquid fit? Expressed in terms of $r$, what is the volume of the cone? of the empty cylinder? of the hemisphere?
Q1. If you shift a stack of cards, does its volume change?
Think of thin layers of water. (Not only helps you to solve the problem, but also to relax!)
In the previous problem, we sliced two 3d shapes with parallel planes and noticed that the resulting cross sections always have equal areas. Our observation led us to claim that the shapes themselves have equal volumes. This line of reasoning, known as Cavalieri's principle, is the mathematical rendition of a simple thought: equal parts make equal wholes.
The method bears the name of Bonaventura Cavalieri (1598-1647) even though he was not nearly the first to employ it. The Ancient Greek Archimedes, and later the Chinese father-son duo of Zu Chongzhi and Zu Gengzhi, used the same method to compute the volume of a sphere (as you will in a moment) more than a thousand years before Bonaventura was born.
In fact, you yourself probably apply Cavalieri's principle more often than you think. For example, every time you compute the volume of a pyramid or a cone (just like now!) you are indirectly referencing the slice argument.
For more delightful examples of this thinking tool, check out a past geometry club problem set!
247. (Continuation) Show that a sphere of radius $r$ encloses a volume of $\frac{4}{3}\pi r^3.$
Q1. How is the volume of a sphere related to the volume of a hemishpere?
Get some practice with the formula here!
The prefix hemi- in hemisphere comes from the ancient greek ἡμι- meaning half. This prefix is related to the Latin semi-, which appears in words like semicircle, semifinal, and semicolon. (From Wiktionary.)
248. Three softball teams ordered equipment from the
same catalog. The first team spent $\$ 285$ on 5 shirts, 4
caps, and 8 bats. The second team spent $\$ 210$ on 12 shirts
and 6 caps. The third team spent $\$ 250$ for 7 shirts, 10
caps, and 3 bats. What were the catalog prices for shirts, caps, and
bats?
Q1. How many variables should you consider? What would you call them?
Q2. Which equation would you start with?
Q3. Can you think of a visual representation of this problem? Think graphs!
Write a system of equations to represent the problem conditions.
Begin solving for your variables by picking the simplest equation.
Express everything in terms of a single variable.
Here's a neat way to visualize this problem: let $x$ represent the price of a shirt, $y$ represent the price of a cap, and $z$ represent the price of bat. This way, any combination of prices for the three items represents a point in 3d-space with coordinates $(x, y, z)$. Now, express the given conditions as functions in terms of $x$, $y$ and $z$. What are the graphs of these functions shaped like in 3d space?
What are the graphs of these functions shaped like in 3d space?
👀 Reveal answer:
They're planes! Since the true combination of prices should fit all three equations, point $(x,y,z)$ lies on each of the planes determined by the three functions you found above.
Change the equations in the diagram below to see what different starting conditions would look like in 3d:
Sometimes, the intersection of three planes isn't a single point. What are the other possibilities? Which sets of planes produce them? Toggle the diagram to check your ideas!
🫣 Peek at the answer:
A plane, a line, or nothing at all.
One of the equations only uses two variables: what can you say about any plane corresponding to such a function?
🌟 Check yourself:
The plane is perpendicular to one of the coordinate axes – can you explain why?
249. Find two equivalent ways to express the slope
of the vector $[\cos\theta , \sin\theta ]$
Q1. How do you determine a vector's slope?
Q2. What other trig functions do you know?
250. Describe the points on the earth's surface
that are visible to a viewer who is 100 miles above the North Pole.
Q1. For our viewer, what are the furthest visible point(s) on the Earth's surface?
Q2. Should the viewer see further along the Earth in any particular direction?
The Earth's radius measures about 3960 miles – with this in mind, draw a very approximate
diagram of the described situation.
Simplify the problem to two dimensions to create a
manageable diagram. Can you spot a right angle?
A word on perspective: Relative to the tallest things on Earth, 100 miles is A LOT. Mount Everest, the tallest of our peaks, rises only a twentieth of this mamoth height. The highest clouds in the atmosphere–noctilucicents–soar only about 50 miles above the Earth's surface. Yet, 100 miles would be barely noticeable from outer space. The Earth measures almost 80 times as much in diameter and dwarfs the given distance almost beyond perception.
Suppose the viewer mentioned in the problem is an exceedingly tall giraffe Raff standing at the North Pole (in a warm sweater!) From Raff's point of view, which points on the Earth's surface lie on the horizon?
251. Points $P$ and $Q$ on the unit circle are reflected images of each other, using the $y$-axis as a mirror. Suppose that $P$ is described by the angle $\theta$; what angle describes $Q$? In terms of $\theta$, what are the rectangular coordinates of $P$? Simplify the expressions $\cos (180-\theta )$ and $\sin (180-\theta )$ by finding two different ways of writing the rectangular coordinates for $Q$.
Q1. Does your answer change depending on whether $P$ and $Q$ are above or below the $x$-axis?
There's little to say but: when in doubt, draw a unit circle!
252. A conical cup is $64/125$ full of liquid.
What is the ratio of the depth of the liquid to the depth of the cup?
Conical cups appear fuller than cylindrical cups –– explain why.
Q1. How can we determine the volume of a conical cup? Check the glossary for help and explanations.
Q2. Is the liquid in a cylindrical cup form a shape similar to the cup?
Describe the shape of the liquid in the cup.
Explain why the two cones are similar –– that is, why one is a smaller version of the other.
Draw a 2d diagram of the cup, representing the cone with a triangle, and find similar triangles on your diagram.
In the diagram below, all the cups have been set to have the same volume of water, and the water's depth in the cone increases at a constant rate. The orange cone shares its radius with the pink cylinder and has the same volume as the green cylinder. What can you say about the rate at which the water's depth increases in the cylindrical cups?
253. Three tennis balls fit snugly inside a cylindrical can. What percent of the available space inside the can is occupied by the balls?
In problem 247, you derived a formula for the volume of a sphere.
Can you think of a solution that sidesteps computing a tennis ball's volume? (Hint: reference problem 244.)
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