Question

monday september 29
all year:
al runs 8 miles per hour. bob runs 1 mile in 8 minutes. carl runs 2 miles in 15 minutes. dan runs 1/4 mile in 3.5 minutes. who runs at the fastest rate?
yesterday:
set a represents the set of integers. set b represents the set of whole numbers. which venn diagram shows the correct placement of numbers in the sets?
a. venn - diagram with set a containing 0, 6, 20 and set b (inside set a) containing - 7, - 32
c. venn - diagram with set a containing - 2, 10, 46 and set b (inside set a) containing 0
b. venn - diagram with set a containing 0, - 2, - 14 and set b (inside set a) containing 79, 4
d. venn - diagram with set a containing - 20, - 7 and set b (inside set a) containing 0, 9, 81
released staar questions:
triangle hjk is similar to triangle lmn.
two right - angled triangles hjk and lmn are shown with side lengths: hj = 4.5 cm, hk = 7.5 cm, jk = 6 cm, lm = 9 cm, ln = 11.25 cm, mn =?
which proportion can be used to calculate the length of mn in centimeters?
a $\frac{7.5}{11.25}=\frac{ln}{4.5}$
b $\frac{9}{7.5}=\frac{lm}{9}$
c $\frac{11.25}{ln}=\frac{4.5}{7.5}$
d $\frac{9}{lm}=\frac{6}{4.5}$

Explanation:

Step1: Convert Bob's rate to miles per hour

Since Bob runs 1 mile in 8 minutes, and there are 60 minutes in an hour. Let $r_{Bob}$ be Bob's rate. We use the proportion $\frac{1}{8}=\frac{r_{Bob}}{60}$, then $r_{Bob}=\frac{60}{8} = 7.5$ miles per hour.

Step2: Convert Carl's rate to miles per hour

Carl runs 2 miles in 15 minutes. Let $r_{Carl}$ be Carl's rate. Using the proportion $\frac{2}{15}=\frac{r_{Carl}}{60}$, cross - multiply gives $15r_{Carl}=2\times60$, so $r_{Carl}=\frac{2\times60}{15}=8$ miles per hour.

Step3: Convert Dan's rate to miles per hour

Dan runs $\frac{1}{4}$ mile in 3.5 minutes. Let $r_{Dan}$ be Dan's rate. Using the proportion $\frac{\frac{1}{4}}{3.5}=\frac{r_{Dan}}{60}$, cross - multiply: $3.5r_{Dan}=\frac{1}{4}\times60 = 15$, then $r_{Dan}=\frac{15}{3.5}=\frac{15\times2}{7}=\frac{30}{7}\approx4.29$ miles per hour.

Step4: Compare the rates

Al's rate $r_{Al} = 8$ miles per hour, Bob's rate $r_{Bob}=7.5$ miles per hour, Carl's rate $r_{Carl}=8$ miles per hour, Dan's rate $r_{Dan}\approx4.29$ miles per hour. Al and Carl have the highest rate among them.

For the set - theory question:
Whole numbers are non - negative integers. Set A (integers) includes positive and negative whole numbers and zero, while set B (whole numbers) includes zero and positive integers.

Step1: Analyze the sets

Integers $\mathbb{Z}=\{\cdots,- 2,-1,0,1,2,\cdots\}$ and whole numbers $\mathbb{W}=\{0,1,2,\cdots\}$. So set B is a subset of set A.

Step2: Check the Venn - diagrams

Option D shows set B (whole numbers: 0, 9, 81) as a subset of set A (integers: - 20, - 7, along with the whole numbers in set B).

For the similar - triangles question:
If $\triangle HJK\sim\triangle LMN$, then the ratios of corresponding sides are equal.
The ratio of the sides of similar triangles gives $\frac{HJ}{LM}=\frac{JK}{MN}=\frac{HK}{LN}$.
We have $HJ = 4.5$, $JK = 6$, $HK = 7.5$, $MN=9$, $LN = 11.25$.
$\frac{HJ}{LM}=\frac{4.5}{LM}$, $\frac{JK}{MN}=\frac{6}{9}$, $\frac{HK}{LN}=\frac{7.5}{11.25}$.
Since $\frac{HK}{LN}=\frac{7.5}{11.25}=\frac{LN}{4.5}$ is incorrect. $\frac{HJ}{LM}=\frac{4.5}{LM}$ and $\frac{JK}{MN}=\frac{6}{9}$, and the correct proportion for finding $LM$ is $\frac{HJ}{LM}=\frac{JK}{MN}$, or $\frac{4.5}{LM}=\frac{6}{9}$. Also, from the ratio of corresponding sides, $\frac{JK}{MN}=\frac{6}{9}=\frac{HK}{LN}=\frac{7.5}{11.25}$, and the proportion $\frac{7.5}{11.25}=\frac{LN}{4.5}$ is wrong, $\frac{6}{9}=\frac{LM}{4.5}$ is wrong, $\frac{11.25}{LN}=\frac{4.5}{7.5}$ is wrong, and the correct one is $\frac{6}{9}=\frac{HK}{LN}$ which is equivalent to $\frac{6}{9}=\frac{7.5}{11.25}$.

Answer:

1. Al and Carl run at the fastest rate.
2. D.
3. None of the given options is correct for the similar - triangles proportion to find an unknown side length (the correct proportion for similar triangles $\triangle HJK\sim\triangle LMN$ should be based on the ratio of corresponding sides $\frac{HJ}{LM}=\frac{JK}{MN}=\frac{HK}{LN}$).