Answered

Owen has enough materials to build up to 10 birdhouses in shop class. Each birdhouse needs 12 square feet of wood. The function $m(b) = 12b$ represents the total amount of wood that Owen would need to build $b$ birdhouses. What domain and range are reasonable for the function?


AD: $10 \leq b \leq 12$

R: $0 \leq m(b) \leq 120$


BD: $0 \leq b \leq 10$

R: $0 \leq m(b) \leq 120$


CD: $0 \leq b \leq 10$

R: $12 \leq m(b) \leq 120$


DD: $0 \leq b \leq 120$

R: $0 \leq m(b) \leq 10$

Verified Answer :

  • The domain represents the number of birdhouses, which ranges from 0 to 10, so \(D: 0 \leq b \leq 10\).
  • The range represents the total wood needed, which is found by evaluating the function at the bounds of the domain: \(m(0) = 12(0) = 0\) and \(m(10) = 12(10) = 120\), so \(R: 0 \leq m(b) \leq 120\).
  • The correct option is the second one: \(\boxed{\text{D: } 0 \leq b \leq 10, \text{ R: } 0 \leq m(b) \leq 120}\).

Explanation

  1. Determining the Domain

The problem asks us to determine the domain and range of a function representing the amount of wood needed to build birdhouses. The variable \(b\) represents the number of birdhouses Owen can build. Since he can build anywhere from 0 to 10 birdhouses, the input variable \(b\) must be in the interval \(0 \leq b \leq 10\). This interval defines the domain of the function. Each birdhouse requires 12 square feet of wood, and the total amount of wood needed is given by the function \(m(b) = 12b\).

  1. Determining the Range

The range of a function consists of all possible output values. To find the range, we evaluate the function at the boundaries of its domain. The minimum value of \(b\) is 0, so the minimum amount of wood needed is \(m(0) = 12 \times 0 = 0\). The maximum value of \(b\) is 10, so the maximum amount of wood needed is \(m(10) = 12 \times 10 = 120\). Thus, the output variable \(m(b)\) must be in the interval \(0 \leq m(b) \leq 120\). This interval defines the range of the function.

  1. Final Conclusion and Selection

By comparing our determined domain (\(0 \leq b \leq 10\)) and range (\(0 \leq m(b) \leq 120\)) with the provided options, we see that the second option is correct. The correct choice is:
\(\text{D: } 0 \leq b \leq 10\)
\(\text{R: } 0 \leq m(b) \leq 120\).

Examples

Understanding domain and range is essential for resource planning. For instance, if a bakery makes up to 50 loaves of bread per day using 2 cups of flour per loaf, the function \(f(l) = 2l\) would represent the total flour used. The domain would be \(0 \leq l \leq 50\) and the range would be \(0 \leq f(l) \leq 100\), helping the bakery manager ensure they always have enough inventory.

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