Calculus: the cooling constant of the casserole

Tutoring calculus or differential equations, Newton’s Law of Cooling will surface. The tutor looks at a real-life example.

In yesterday’s post I mention that a casserole dish taken out of the oven cooled from 177C to about 40C during one hour.

Newton’s Law of Cooling can be used to calculate the temp of a cooling object:

Tf = Tiekt

where

Tf = final temp

Ti = initial temp

k = the constant of cooling (if cooling, k will turn out negative)

t = time (usually in seconds)

For this case, we have t=3600 (3600s in one hour):

40 = 177ek3600

Dividing both sides by 177 gives

0.226=e3600k

Now we ln both sides:

ln0.226 = 3600k

Finally we divide by 3600:

-4.13×10-4 = k

Apparently the cooling constant of the casserole is -4.13×10-4.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto: D.C. Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: ideas about infinity: how an infinite series can have a finite sum

Tutoring math, you realize that infinite series can be hard to visualize. The tutor offers perspective.

An infinite series is a sum of terms that never end. An example is a repeating decimal, such as 0.2222222….., which means

0.2222222…. = 0.2 + 0.02 + 0.002 + 0.0002 + 0.00002 + ….

Perhaps surprising, at first, is that a sum may be of infinitely many terms, but have finite value. Yet, most people will agree that

0.2222222…. < 0.3
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: what discount will exactly cancel the tax?

Tutoring high school math, you see financial word problems. The tutor gives an example.

Problem: At 12% tax, what discount will lead to paying exactly the sticker price after tax?

Solution

The final price, f, is as follows:

f = (1-discount)(1+tax)p

Where p is original price, and discount and tax are in decimals, not percent (eg, tax is 0.12 rather than 12%)

We need f=p, at tax=0.12:

p=(1-discount)(1+0.12)p

Dividing both sides by p we get

1=(1-discount)(1.12)

Then, dividing by 1.12,

0.8929 = 1-discount

Rearranging we get

discount = 1-0.8929 = 0.1071 or 10.71%.

Apparently, at 12% tax, a discount of 10.71% is needed to equate the after-tax price to the sticker price.

Cheers.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: how far away is the goose?

Tutoring math, you notice that people like relatable examples. The tutor brings up his observation of a Canada goose.

Looking out over a lake in Nanaimo on Sunday morning, I saw an exceptional Canada goose swimming apart from the others.

The setting was so tranquil, even distance seemed irrelevant: I felt that, on a whim, I could suddenly scoot down to the water, plunge in, and join the goose if I wanted. Yet, how far away was it, really?

Holding my phone at 20cm, or 200mm, I observed the goose at about one-sixth of my phone’s lens port, which is 7mm across. So, to me, the goose appeared 7/6=1.17mm. Yet, a goose that prosperous would likely be about 60cm, or 600mm, from tail to breast. From optics,

object distance/image distance = object length/image length

Therefore,

d/200 = 600/1.17

Multiplying both sides by 200, we get

d=200(600)/1.17 = 102574mm or about 103m

Source:

Bull, John and John Farrand, Jr. The Audubon Society Field Guide to North American Birds, eastern region. New York: Alfred A. Knopf, 1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Excel: entering complex numbers

Tutoring differential equations or complex variables, you might use a spreadsheet sometimes. The tutor gives a hint about entering complex numbers on Excel.

Excel does, indeed, handle complex numbers. Formulas for them are among the Engineering ones.

Apparently, to enter a complex number, quotes are needed around it. For instance, to obtain the product of 6-i and 2+i, you would enter

=improduct(“6-i”,”2+i”)

Hopefully you receive the answer 13+4i.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Excel: square roots of negative numbers

Tutoring college math, complex numbers will likely arise. The tutor mentions using Excel.

Square roots of negative numbers (aka, complex numbers or imaginary numbers) may not be encountered by many high school students. However, in college math they are used, and electricians also use them.

Excel will calculate the square root of a negative number, using the imsqrt() function. Example:

=imsqrt(-9)

gives, on this desktop, 1.8377E-16 + 3i.

1.8377E-16 is so tiny, of course, it’s equivalent to zero. So the effective answer is 3i, which is correct. An electrician might call it j3.

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculator Usage: Euler’s identity with the Nexus 4’s on-board scientific

Tutoring math, calculators can always be interesting. The tutor shares a capability of the Nexus 4 on-board calculator.

If the Nexus 4 on-board calculator has broad complex number capabilities, I don’t know about them. I tried √(-4) and got Not a number.

Yet, behold!

Neat, eh?

Source:

Wikipedia

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: How to convert between fraction and decimal on the Casio fx-991ES PLUS C

Tutoring math, you notice the peculiarities of scientific calculators. The tutor brings up one about the Casio fx-991ES PLUS C.

If you enter the operation 14÷9 into the Casio fx-991ES PLUS C, you’ll just as likely see 14/9 on the screen. If you want the decimal, just press the S⇔D key.

You can keep pressing the S⇔D key to toggle through various formats of the answer.

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Grapes to raisins: the condensation ratio

Tutoring math, all kinds of relationships might call out to you. The tutor brings up the water loss from grape to raisin.

According to one source, fresh grapes contain 1g fibre per 138g. According to another source, raisins contain 1g fibre per 40g.

Therefore, assuming the only loss from grape to raisin is water, 98g of 138g is lost as grapes become raisins. The percentage weight loss is 98/138 = 71%.

Neat, eh?

Source:

www.sun-world.com

no name® Thompson seedless raisins package

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculator usage, math: logs in variable bases with the Casio fx-991ES PLUS C

Tutoring math, you come to appreciate calculators. The tutor points out an interesting feature on the Casio fx-991ES PLUS C.

Typically, the log key on a calculator means “base 10 log”, aka “common log.” What about base 7 log, for instance?

The Casio fx-991ES PLUS C has a log□ key which can accept an input for the base.

Example: On the Casio fx-991ES PLUS C, evaluate log762.

Solution:

  1. Press the log□ key, second down on the right.
  2. Press 7, then arrow over once, then key 62, then arrow out of the brackets.
  3. Press =

Everyone not blessed with the Casio fx-991ES PLUS C can just enter log62/log7 =

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.