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.
a. Use the Distributive Property to rewrite the bear population in Year 3
without needing to use parentheses.
Jin's expression for the number of wolves in Year 4 is (x+a)(x-b), where a and b are whole numbers greater than 1 and less than 10.
One possible expression that Jin could use is (x+3)(x-2).
Now, let's multiply the two terms of the expression:
(x+3)(x-2) = x*x + x*(-2) + 3*x + 3*(-2)
= x^2 - 2x + 3x - 6
= x^2 + x - 6
Therefore, the expression Jin uses to describe the number of wolves in the park during Year 4 of the study could be x^2 + x - 6.
To rewrite the bear population in Year 3 without parentheses, we can use the Distributive Property to simplify the expression 2(x+10).
2(x+10) = 2x + 2(10)
2(x+10) = 2x + 20
Therefore, the bear population in Year 3 can be written as 2x + 20 bears.
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. Explain the answer.
Let's begin by writing an equation to represent the total number of bears in the park during Year 2. We know that in Year 2, the population of bears is x + 10. Given that the total number of bears in Year 2 is 42, we can write the equation:
x + 10 = 42
Now, we can solve this equation to find the value of x, which represents the number of bears in the park during Year 1.
Subtracting 10 from both sides of the equation:
x = 42 - 10
x = 32
Therefore, there were 32 bears in the park during Year 1. This is consistent with the information provided because the population increased by 10 bears in Year 2 (making it x + 10 = 32 + 10 = 42 bears).
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.
Let's write an equation to represent the total number of bears in the park during Year 3. We know that in Year 3, the population of bears is 2(x + 10). Given that the total number of bears in Year 3 is 50, we can write the equation:
2(x + 10) = 50
Now, we can solve this equation to find the value of x, which represents the number of bears in the park during Year 1.
Expanding the left side of the equation using the Distributive Property:
2x + 20 = 50
Subtracting 20 from both sides of the equation:
2x = 30
Dividing by 2 to solve for x:
x = 15
Therefore, there were 15 bears in the park during Year 1. This is different from the 32 bears calculated in part b because the population growth rate from Year 1 to Year 2 was not the same as the growth rate from Year 2 to Year 3.
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.
To solve the inequality y + 5 > 38, we first subtract 5 from both sides:
y + 5 - 5 > 38 - 5
y > 33
Therefore, the possible values for y are any number greater than 33. Here are three possible values for y: 34, 40, 50.
Jin is a wildlife researcher hired to track the population of wolves in the same
national park. In Year 1 of the study, there are z wolves; in Year 2 of the
study, there are 7+z wolves; and in Year 3 of the study, there are 3z-3 wolves.
The number of wolves in Year 2 is equal to the number of wolves in Year 3.
a. Write and solve a linear equation to determine the value of z
Let's first write an equation based on the information given:
The number of wolves in Year 2 is 7+z.
The number of wolves in Year 3 is 3z-3.
According to the given information, the number of wolves in Year 2 is equal to the number of wolves in Year 3:
7+z = 3z - 3
Now we can solve this equation for z:
Add 3 to both sides:
7 + z + 3 = 3z - 3 + 3
z + 10 = 3z
Subtract z from both sides:
z + 10 - z = 3z - z
10 = 2z
Divide by 2:
10/2 = 2z/2
5 = z
Therefore, the value of z is 5.