Question

1. a bulldozer clears a rectangular piece of land of sides 1.732 km × 1.5 km every week. find the area the bulldozer can clear in 100 days, leave the answer in standard form.
2. i am a 7 - digit number, my median is 5, my second and third digits are consecutive odd integers whose sum is 8. my first digit is 1 less than the sum of my second and third digits. the sum of my first and last digits is 9, my sixth digit is 3 more than my last digit and my fifth digit is 4 less than my median. who am i? write the answer in standard form.
3. i am a 6 - digit number, my first digit is 3 more than my second digit, my third digit is the sum of my first and second digits, my fourth digit is 1 less than my third digit, my fifth digit is twice my second digit and my sixth digit is 2 less than my fifth digit. if the average of my third and fourth digits is 6\frac{1}{2}, find the value for each of the 6 - digits. write the 6 - digit number in words and express the answer in standard form.

Explanation:

Step1: Solve problem 8

• First, find the weekly - cleared area.

The area of a rectangle is $A = l\times w$, where $l = 1.732$ km and $w = 1.5$ km. So, $A_{weekly}=1.732\times1.5 = 2.598$ km².

• Then, find the daily - cleared area.

There are 7 days in a week, so $A_{daily}=\frac{2.598}{7}=0.3711428571$ km².

• Finally, find the area cleared in 100 days.

$A_{100 - days}=0.3711428571\times100 = 37.11428571\approx3.711428571\times10^{1}$ km².

Step2: Solve problem 9

• Let the 7 - digit number be $abcdefg$.

Since the median of a 7 - digit number (in ascending or descending order) is the 4th digit, $d = 5$.
Let the second and third digits be $x$ and $y$. Since $x$ and $y$ are consecutive odd integers and $x + y=8$, we have $x = 3$ and $y = 5$.
The first digit $a=(x + y)-1=7$.
Since $a + g = 9$ and $a = 7$, then $g = 2$.
The sixth digit $f=g + 3=5$.
The fifth digit $e=d - 4=1$.
So the 7 - digit number is $7355152$.

Step3: Solve problem 10

• Let the second digit be $x$. Then the first digit $a=x + 3$.

The third digit $c=a + x=2x + 3$.
The fourth digit $d=c - 1=2x+2$.
The fifth digit $e = 2x$.
The sixth digit $f=e - 2=2x - 2$.
The average of the third and fourth digits is $\frac{c + d}{2}=\frac{(2x + 3)+(2x + 2)}{2}=\frac{4x + 5}{2}$.
Since $\frac{4x + 5}{2}=6.5$, we solve for $x$:
\[

$$\begin{align*} \frac{4x + 5}{2}&=6.5\\ 4x+5&=13\\ 4x&=8\\ x&=2 \end{align*}$$

\]
So the first digit $a=x + 3=5$, the second digit $x = 2$, the third digit $c=2x + 3=7$, the fourth digit $d=2x + 2=6$, the fifth digit $e = 4$, the sixth digit $f=2x - 2=2$.
The 6 - digit number is $527642$.

Answer:

• Problem 8: $3.711428571\times10^{1}$ km²
• Problem 9: $7355152$
• Problem 10: $527642$