Substitute $$n = 5$$: $$4(5)^2 - 3(5) + 7 = 4(25) - 15 + 7 = 100 - 15 + 7 = 92$$ tiles.

Substitute $$w = 7$$: $$2(3(7) + 4) + 2(2(7) - 1) = 2(21 + 4) + 2(14 - 1) = 2(25) + 2(13) = 50 + 26 = 76$$ feet.

When you encounter an expression with variables and need to find its value for a specific number, you're working with substitution and order of operations. The key is to replace the variable with the given value and then carefully follow the order of operations (PEMDAS).

Let's substitute $$x = 4$$ into the expression $$5 + 3(x - 2)^2$$:

$$5 + 3(4 - 2)^2$$

Now follow PEMDAS step by step. First, handle the parentheses: $$4 - 2 = 2$$, giving us $$5 + 3(2)^2$$.

Next, calculate the exponent: $$(2)^2 = 4$$, so we have $$5 + 3(4)$$.

Then multiply: $$3 \times 4 = 12$$, leaving us with $$5 + 12$$.

Finally, add: $$5 + 12 = 17$$. The answer is C.

Let's see where the wrong answers come from. Choice A (41) likely results from incorrectly calculating $$(4-2)^2$$ as $$4^2 - 2^2 = 16 - 4 = 12$$, then doing $$5 + 3(12) = 41$$. Choice B (29) probably comes from substituting first to get $$5 + 3(4-2)^2$$, but then mistakenly calculating $$3 \times 4 - 2 = 12 - 2 = 10$$, squared to get $$100$$, then somehow getting 29. Choice D (21) might result from forgetting to square the 2, calculating $$5 + 3(4-2) = 5 + 3(2) = 5 + 6 = 11$$, or other order of operations errors.

Always substitute first, then follow PEMDAS carefully. Write out each step to avoid rushing through the order of operations.

Substitute $$w = 7$$ into the expression: $$45 + 12(7) - 30 = 45 + 84 - 30 = 129 - 30 = 99$$.

Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=4 in expression 3x+5, write 3(4)+5 replacing x with 4, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: 3x+5 at x=4 becomes 3(4)+5, multiply first: 12, then add: 12+5=17. For this specific question, evaluate 3x+5 at x=4: substitute 3(4)+5, evaluate 3×4=12 (multiply), then 12+5=17 (add). Correct substitution and evaluation yield 17, which matches choice C. Common errors include concatenation (3x with x=4 as 34 instead of 3×4=12) or violating order by adding before multiplying (3+5=8, then 8×4=32). Process: (1) write expression 3x+5, (2) substitute x=4 to get 3(4)+5, (3) evaluate using order (multiply then add to get 17), (4) verify reasonable (3×4=12, +5=17 makes sense).

When you encounter a problem involving a mathematical model or formula, your job is to substitute the given values and carefully perform the calculations step by step.

Here, you need to find the profit when $$h = 2$$ (since 2:00 PM is 2 hours past noon). Substitute $$h = 2$$ into the profit formula $$150 + 25h - 5h^2$$:

$$150 + 25(2) - 5(2)^2$$

Work through this systematically: $$25(2) = 50$$ and $$5(2)^2 = 5(4) = 20$$. So you have:

$$150 + 50 - 20 = 180$$

The profit at 2:00 PM is $180, which is answer choice C.

Let's see where the wrong answers might come from. Choice A ($190) likely results from forgetting to subtract the $$5h^2$$ term entirely, giving you $$150 + 50 = 200$$, or making an arithmetic error. Choice B ($170) might occur if you miscalculate $$5(2)^2$$ as 10 instead of 20, giving you $$150 + 50 - 10 = 190$$, or make another computational mistake. Choice D ($160) could result from incorrectly calculating $$25(2)$$ as 30 instead of 50, leading to $$150 + 30 - 20 = 160$$.

Remember: when substituting into algebraic expressions, follow the order of operations carefully. Calculate exponents first, then multiplication, and finally addition and subtraction from left to right. Double-check each arithmetic step to avoid careless errors.

Substitute $$t = 3$$: $$T = 20 + 5(3) - 2(3)^2 = 20 + 15 - 2(9) = 20 + 15 - 18 = 17$$°C. Choice B results from calculating $$20 + 5(3) - 2(3) = 20 + 15 - 6 = 29$$, not squaring the 3. Choice C results from $$20 + 5(3) = 35$$, forgetting the $$-2t^2$$ term. Choice D results from $$20 + 5(3) + 2(3)^2 = 53$$, adding instead of subtracting the $$2t^2$$ term.

Substitute $$x = 4$$: $$2(4)^2 + 5(4) - 3 = 2(16) + 20 - 3 = 32 + 20 - 3 = 49$$.

Substitute the values: $$\frac{2(6) + 9}{3} - 6 = \frac{12 + 9}{3} - 6 = \frac{21}{3} - 6 = 7 - 6 = 1$$.

When you see an expression with variables like $$25 + 0.50m$$, you're working with substitution – replacing the variable with a given number and calculating the result. This type of problem tests whether you can follow the order of operations while substituting values.

To find the total cost when someone drives 84 miles, substitute $$m = 84$$ into the expression $$25 + 0.50m$$:

$$25 + 0.50(84)$$

First, multiply: $$0.50 \times 84 = 42$$

Then add: $$25 + 42 = 67$$

The total cost is $67, making B correct.

Let's examine why the other answers are wrong. Choice A ($59) likely comes from incorrectly calculating $$0.50 \times 84$$ as $34 instead of $42, then adding $$25 + 34 = 59$$. Choice C ($42) is the result of only calculating the mileage portion ($$0.50 \times 84 = 42$$) but forgetting to add the base fee of $25. Choice D ($109) probably results from adding $$25 + 84 = 109$$, completely ignoring the 0.50 multiplier – this shows misunderstanding of how the expression works.

When substituting into expressions, always follow order of operations: handle multiplication before addition, and make sure you substitute the variable everywhere it appears. Double-check that you're using all parts of the original expression in your final calculation.