Roger is a wildlife researcher who has been hired to study and track the population of bears in a national park. In Year 1 of the study, Roger determines that the park is home to x

Question

Roger is a wildlife researcher who has been hired to study and track the population of bears in a national park. In Year 1 of the study, Roger determines that the park is home to  x
  bears. In Year 2, the population has increased to  x+10
  bears, and in Year 3, the population has increased to  2(x+10)
  bears.
Use the Distributive Property to rewrite the bear population in Year 3 without needing to use parentheses.
The total number of bears in the park during Year 2 is 42. Write and solve a linear equation to determine how many bears were in the park during Year 1.
 The total number of bears in the park during Year 3 is 50. Write and solve a linear equation to determine how many bears were in the park during Year 1. Note that this answer will not be the same as the answer you got in part  b
 .
During Year 4, bad weather keeps Roger from finding the exact number of bears in the park. However, he knows that  y+5>38
 , where  y
  is the number of bears in the park during Year 4. Solve the inequality, and give three possible values for  y
 .

GPT-4o mini · Answer

**Step 1: Rewrite the Year 3 Population Using the Distributive Property** The population of bears in Year 3 is given as $ 2(x + 10) $. Using the Distributive Property: $$ 2(x + 10) = 2x + 20. $$ So, the bear population in Year 3 without parentheses is $ 2x + 20 $. --- **Step 2: Determine the Number of Bears in Year 1 Based on Year 2 Population** We know that the total number of bears in Year 2 is 42: $$ x + 10 = 42. $$ To solve for $ x $: $$ x = 42 - 10, $$ $$ x = 32. $$ So, the number of bears in Year 1 is **32**. --- **Step 3: Determine the Number of Bears in Year 1 Based on Year 3 Population** We know that the total number of bears in Year 3 is 50: $$ 2(x + 10) = 50. $$ First, we can simplify the equation: $$ 2x + 20 = 50. $$ Now, solving for $ x $: $$ 2x = 50 - 20, $$ $$ 2x = 30, $$ $$ x = 15. $$ So, the number of bears in Year 1 is **15**. --- **Step 4: Solve the Inequality for Year 4** We know that $ y + 5 > 38 $. To solve for $ y $: $$ y > 38 - 5, $$ $$ y > 33. $$ Possible values for $ y $ can be any numbers greater than 33. Therefore, three possible values for $ y $ could be: - 34 - 35 - 36 Thus, values for $ y $ could be $ 34, 35, $ or $ 36 $.