1.5E Exercises
- Page ID
- 152882
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. According to College Board at the time of writing this, the average cost per year to attend Siena College, after aid, is $32,810. In Fall 2022, Siena College had a total undergraduate enrollment of 3,519 students. Assuming everything stays about the same going forward, estimate how much money Siena will receive each year from its students' payments.
2. One of my washing machine models is sold for the low low price of $789.99. If I sold 32 of them last month, about how much revenue did I take in?
3. Approximately 10% of people are left-handed. About 19.62 million people live in the New York-Newark-Jersey City metropolitan area. Roughly how many of them are left-handed?
4. Approximately 2% of people are redheads (in the northern hemisphere, anyway). Roughly how many redheads are there in the metropolitan area?
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1. We just need to multiply the average amount by the total number of students. 32810 is pretty close to 33000 and 3519 is pretty close to 3500. With quick scratch work, \( 33 \times 35 = 1155 \), to which I tack on a total of five zeros to get $115,500,000. With a calculator, \( 32810 \times 3519 = $115,458,390\).
2. 32 is about 30, and the price is about $800 each. I estimate about $24,000 in revenue, and I could even say hey, the other 2 washers on top of that would bring in about $1600 as well. All together, I estimate something between $24,000 and $25,600. A calculator gives the exact answer $25,279.68.
2. 19.62 million means 19,620,000 people. The decimal version of 10% is 0.1, so we just need to multiply the two numbers. We use the trick of moving the invisible decimal one place to the left, to get 1,962,000.0 people. Dang, almost two million lefties!
4. Convert from 2% to \(0.02\). That's double \(0.01\), so let's think about that first. We can find \(0.01(19,620,000) = 196200.00\) by moving the decimal point. So 1% is 196,200 people, which I could round up to 200,000ish and finally double. I expect a bit under 400,000 redheads.
Mohammad bought a box of dates in preparation for Ramadan. The box's dimensions are 2 inches by 6 inches by 10 inches. On the back, it a serving size is 2 dates and there are about 35 servings per container. Is that a reasonable claim?
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If these are some big ol' Medjool dates, each one might have a volume of 2 cubic inches (approximating their oblong shape with a 1x1x2 inch rectangular prism). The total volume of the box is \(2\cdot 6 \cdot 10 = 120\) cubic inches. The question is, can a total of 70 dates fit in that box? If we divide the total volume by the estimated volume of a date, we expect 60 dates to fit in. But dates are squishy and not actually rectangular, so it seems reasonable that an extra 10 could be crammed in, especially if some are smaller than others. It's probably not false advertising!
Driving home just now, I heard a guy on the radio claim that "the average American now spends $5000 a year on gas." I was immediately skeptical because I feel pretty average and I've never spent remotely that much on gas per year. I started doing some mental math to entertain myself while driving and evaluate this claim.
- Assuming that it costs $50 on average to fill up your gas tank, how many times a year would you have to fill up to reach a total of $5000?
- Filling up that many times, how many times a week would you be getting gas (and not just topping up, a full tank!)?
- Conservatively assuming I get 20 miles to the gallon on average (both cars that I've owned in my life did much better, in fact), how many miles a week would I be driving to use that much gas?
- There are 52 weeks in a year, so even rounding down to 50 weeks, how many miles driven per year would that be?
- What possible motivations could someone have for exaggerating the amount of money Americans spend on gas per year?
I now believe that the radio guy meant to say that the average HOUSEHOLD spends $5000 a year on gas. That is certainly more believable! But a quick search showed that the $5000/household figure comes from 2022, and the guy is announcing it in late 2024? Meanwhile household gasoline yearly costs have been dropping between 2022 and 2024, possibly even by about half!
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1. That's 100 fill-ups per year.
2. Roughly twice a week! Meanwhile, I think I have pretty normal driving habits and I only fill up maybe two to three per month...
3. Say that filling up twice a week is buying about 25 gallons of gas. If I'm using 25 gallons at 20 miles per gallon, I must be driving 500 miles a week.
4. That would be putting 25,000 miles a year on my car. I went home and looked up the average miles driven per year, and it's actually only about 15,000.
5. Hmmm, makes ya wonder... Maybe someone wants to make current gas prices sound like much more of a crisis than they are... Maybe someone wants to influence more people to buy electric cars... What do you think?
Without actually evaluating anything or using a calculator, which is bigger?
- \( \frac{6}{51} \) vs \(\frac{6}{49}\)
- \( \frac{7}{11}\) vs \(\frac{8}{11} \)
- \( \frac{5}{4} \) vs \( \frac{9}{7} \)
- \( 3\pi \) vs \(9\)
- \( 6 \) vs \(2e\)
- \( \sqrt{8} \) vs \( \sqrt{7}\)
- \( 5 \) vs \( \sqrt{29} \)
- \( \sqrt{101} \) vs \( \frac{48}{5} \)
- \( \frac{1}{2} \) vs \( \left(\frac{1}{2}\right)^2 \)
- \( \frac{1}{n} \) vs \( \frac{1}{n+1} \)
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- \(\frac{6}{49}\)
- \(\frac{8}{11} \)
- \( \frac{9}{7} \) (Hint: get a common denominator.)
- \( 3\pi \)
- 6
- \( \sqrt{8} \)
- \( \sqrt{29} \)
- Notice that \( \sqrt{101} \) must be a tad more than 10, but \(\frac{48}{5} \) is a bit less than \( \frac{50}{5}=10\). So \( \sqrt{101} \) is bigger.
- \( \frac{1}{2} \) is bigger than \( \frac{1}{4}\). This is important! If you take a lil fraction less than 1 and raise it to a positive integer power, the result will get smaller! But with numbers bigger than 1, raising them to a positive integer power makes a bigger number.
- The denominator \( n+1 \) is bigger than \(n\), so its fraction is a smaller number. Don't believe me? Pick \(n = 2\) for example and see what happens.
The only benefit I see to the practice of rationalizing denominators is because in some cases it's easier to grasp the rough size of a number. For example, I don't look at \( \frac{1}{\sqrt{2}}\) and immediately get a sense of how much it is. But if you rationalize the denominator, it becomes much easier to estimate. Give it a shot.
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By writing \( \frac{1}{\sqrt{2}} \cdot \frac{ \sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\), it becomes clear that this is half of the number \(\sqrt{2}\). I've seen that \(\sqrt{2} \approx 1.414\)ish, and half of that is \(0.7\)ish. That wasn't immediately clear to me from the \( \frac{1}{\sqrt{2}}\) version.
I'm selling two washing machine models for the same retail price. With Model A, I mark it as 30% off on Monday, and then on Tuesday, I mark it down again, 15% off the sale price! Meanwhile, with Model B, I put it on sale for 15% off on Monday, and then mark it down again Tuesday, 30% off the sale price! Which one is the better deal?
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Maybe you realize that when you calculate percentages, 50% of a larger number is bigger than 50% of a smaller number. Perhaps you thought, well if I take off 30% first on Model A, that's a lot of money, compared to taking 30% off a lower price on Model B... Or maybe you had another thought! Either way, let's shine the inexorable light of math on the situation.
Whatever the original price is, let's call it \(x\). For Model A, the price on Monday has 30% removed, so it's \( (1-0.3)x = 0.7x \), and on Tuesday there is an additional markdown of 15%. That gives an ultimate sale price of \( (1-0.15)(0.7x) = 0.85 \cdot 0.7 \cdot x\). For Model B, we start with \(x\) and mark it down by 15%, \( (1-0.15)x = 0.85 x \), and then take the new price and take off 30%, to get \( (1-0.3)(0.85 x) = 0.7 \cdot 0.85 \cdot x \). But wait, multiplication is commutative, so it doesn't matter what order you do it in! Those final prices are exactly the same!