Question

1. the perimeter of certain soccer field is 310 m. the length is 65 m more than the width. what are the dimensions?

perform the following operations for 21.22.

1. \\(\frac{0}{3a - 2a}=\\) \t\t\t 22) \\(\frac{4x - 5}{0}=\\)

multiply binomials for # 23 and # 24.

1. \\((x + 3)(2y - 2)\\) \t\t\t 24) \\((x - 2y)(x - 2y)\\)

simplify expressions, using exponential properties for # 25.26

1. \\((3ab)(2b)\\) \t\t\t 26) \\((-2y)^2(5y)\\)
2. fifty subtracted from ten times a number is 90. find the number.
3. solve \\(5x - 10 = 3x + 2\\)
4. a 7.5 % raise brings a salary to $ 46,225. what was a salary before the raise?
5. light travels about 9,460,000,000,000 km in one year. write scientific notation for this number.

Explanation:

20) Step1: Define variables

Let width = $w$, length = $w+65$

20) Step2: Use perimeter formula

Perimeter $P=2(l+w)$, so:
$$310=2((w+65)+w)$$

20) Step3: Simplify and solve for $w$

$$310=2(2w+65)$$
$$155=2w+65$$
$$2w=155-65=90$$
$$w=45$$

20) Step4: Find length

$l=45+65=110$

---

21) Step1: Simplify denominator

$3a-2a=a$

21) Step2: Evaluate fraction

$\frac{0}{a}=0$ (for $a
eq0$)

---

22) Step1: Identify undefined case

Division by 0 is undefined.

---

23) Step1: Apply distributive property

$$(x+3)(2y-2)=x(2y-2)+3(2y-2)$$

23) Step2: Expand and simplify

$$=2xy-2x+6y-6$$

---

24) Step1: Rewrite as square

$$(x-2y)(x-2y)=(x-2y)^2$$

24) Step2: Expand the square

$$=x^2-4xy+4y^2$$

---

25) Step1: Multiply coefficients and variables

$$(3ab)(2b)=3\times2\times a\times b\times b$$

25) Step2: Simplify using exponents

$$=6ab^2$$

---

26) Step1: Simplify the squared term

$$(-2y)^2=4y^2$$

26) Step2: Multiply by remaining term

$$4y^2(5y)=20y^3$$

---

27) Step1: Define variable and set equation

Let number = $n$, so:
$$10n-50=90$$

27) Step2: Solve for $n$

$$10n=90+50=140$$
$$n=14$$

---

28) Step1: Isolate $x$ terms

$$5x-3x=2+10$$

28) Step2: Solve for $x$

$$2x=12$$
$$x=6$$

---

29) Step1: Define original salary

Let original salary = $S$, new salary = $1.075S$

29) Step2: Solve for $S$

$$S=\frac{46225}{1.075}=43000$$

---

30) Step1: Rewrite in scientific notation

Move decimal 12 places left:
$$9.46\times10^{12}$$

Answer:

1. Width = 45 m, Length = 110 m
2. $0$
3. Undefined (division by zero)
4. $2xy-2x+6y-6$
5. $x^2-4xy+4y^2$
6. $6ab^2$
7. $20y^3$
8. $14$
9. $x=6$
10. $\$43,000$
11. $9.46\times10^{12}$ km