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CAHSEE Math 4.5 Measurement and Geometry 181 Views


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Description:

Measurement and Geometry Drill 4 Problem 5. What is the surface area in centimeters squared?

Language:
English Language

Transcript

00:03

Here’s your shmoop du jour...

00:07

The volume of Wyatt’s perfectly spherical head is 36 pi inches cubed.

00:12

What is its surface area in centimeters squared?

00:15

For the sake of the problem, we can just imagine that Wyatt is sans ears or nose.

00:24

And here are the potential answers...

00:29

So… the question LOOKS like a beast –…

00:31

but it’s actually just a really simple conversion one – from inches to centimeters.

00:35

First some translation work is needed in calculating the surface area.

00:39

The key ingredient we need here is r, or radius, which we can then plug in to the formula which

00:45

they give us as 4 pi r squared.

00:48

We can plug in 36 pi into the formula to get

00:51

36 pi equals 4 thirds pi r-cubed.

00:56

We simplify and solve for r.

00:59

So… we multiply both sides by 3 fourths to get rid of that 4 thirds thing

01:07

and we have 27 pi equals pi r cubed.

01:11

Then we just divide both sides by pi to get 27 equals r cubed.

01:16

That 27 is kind of a magic number and should always trigger that green light in your brain

01:20

that it’s the cube of 3.

01:27

In this case, it’s no exception… because that’s exactly what we’re after.

01:30

R cubed is 27, so r is 3.

01:34

Now we can plug the 3 into the formula they give us

01:37

for the surface area of a sphere and that gets written as:

01:40

Area is 4 pi r squared

01:43

Or 4 pi 3 squared

01:46

Or 4 pi times 9

01:49

Or 4 times 9 here is 36.

01:52

36 pi....inches squared.

01:55

Now we just convert from inches to centimeters and we’re home free.

01:58

BUT WAIT, there’s more! A dirty little trick at play here.

02:03

Surface area is in inches SQUARED, not just inches.

02:07

They give us that one inch is 2.5 centimeters ...but 1 SQUARE inch doesn't equal 2.5 centimeters...

02:14

…it equals the quantity 2.5 centimeters SQUARED.

02:18

2.5 squared equals 6.25. So 1 square inch equals 6.25 centimeters squared.

02:25

We can then take 36 pi inches squared and multiply it by 6.25, which equals 225 pi square centimeters.

02:34

Looks like our answer is D!

02:36

As in “Dome head.”

02:37

Another way we could've done this is converted the initial radius of 3 inches to centimeters

02:42

first and then plugged in the converted radius into the surface area formula.

02:46

1 inch equals 2.5 centimeters, so the radius is approximately 7.5 centimeters, or 15 over 2.

02:54

Surface area equals 4 pi times 15 over 2 squared.

02:59

15 over 2 squared equals 225 over 4.

03:06

The 4's cancel out on top and bottom, so the area equals 225 pi centimeters squared,

03:12

just like the answer we got above.

03:14

Now that we know the answer, it’s seems as plain as the nose on Wyatt’s face.

03:17

Oh. Sorry, Wyatt.

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