Question

grade 7 lesson 8 solve problems with rational numbers

1 to the nearest year, what is the difference in antoine’s age and katy’s age?
a antoine is about 17 years older than katy.
b antoine is about 6 years older than katy.
c antoine is about 6 years younger than katy.
d antoine is about 17 years younger than katy.

2 a scientist measures the change in the temperature of a chemical solution. at the start of the experiment, the solution is at a temperature of −14.96°c. the temperature decreases 2.9°c each hour.
part a
write an expression that can be used to find the temperature of the solution after 3 hours.
answer: ______
part b
what is the temperature of the solution after 3 hours, rounded to the nearest whole degree? estimate to show your answer is reasonable.
show your work.
answer: ______°c

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Explanation:

Part A

Step1: Define initial temperature and rate

Let the initial temperature be \( T_0 = -14.96^\circ\text{C} \) and the rate of temperature decrease be \( r = -2.9^\circ\text{C per hour} \) (negative because it's a decrease). The time is \( t = 3 \) hours. The formula for the temperature after \( t \) hours is \( T = T_0 + r\times t \).

Step2: Write the expression

Substituting the values, the expression is \( T = -14.96 + (-2.9)\times 3 \) or simplified as \( T = -14.96 - 2.9\times 3 \).

Step1: Calculate the total decrease

First, find the total temperature decrease in 3 hours. The rate is \( 2.9^\circ\text{C per hour} \), so in 3 hours, the decrease is \( 2.9\times 3 = 8.7^\circ\text{C} \).

Step2: Find the final temperature

The initial temperature is \( -14.96^\circ\text{C} \). Subtract the total decrease (since it's a decrease, we add a negative value) : \( -14.96 - 8.7 = -23.66^\circ\text{C} \).

Step3: Round to nearest whole number

Rounding \( -23.66 \) to the nearest whole number, we look at the tenths place (6), which is greater than or equal to 5, so we round down the ones place: \( -24^\circ\text{C} \).

Answer:

\( -14.96 - 2.9\times 3 \) (or equivalent expression like \( -14.96 + 3\times(-2.9) \))

Part B