To solve this problem, let’s go through the details step-by-step:
1. Identifying the Inequality:
Joe needs a minimum of 50 total rooms. He has already reserved 16 rooms, so he needs additional rooms to reach or exceed the total required. Each additional block of rooms contains 8 rooms.
Let [tex]\( B \)[/tex] represent the number of additional blocks Joe reserves.
The total number of rooms Joe will have after reserving [tex]\( B \)[/tex] additional blocks is [tex]\( 16 + 8B \)[/tex].
Joe needs this total to be at least 50. This can be represented by the inequality:
[tex]\[
16 + 8B \geq 50
\][/tex]
Therefore, the correct inequality is:
[tex]\[
(B) \quad 16 + 8B \geq 50
\][/tex]
2. Determining the Additional Cost:
Joe needs to determine the least number of additional blocks required to meet the room requirement. We start by solving the inequality:
[tex]\[
16 + 8B \geq 50
\][/tex]
Subtract 16 from both sides:
[tex]\[
8B \geq 34
\][/tex]
Divide both sides by 8:
[tex]\[
B \geq \frac{34}{8}
\][/tex]
[tex]\[
B \geq 4.25
\][/tex]
Since [tex]\( B \)[/tex] represents the number of whole blocks (partial blocks are not possible), Joe will need to round up to the next whole number. Therefore,
[tex]\[
B = 5
\][/tex]
Each block costs \[tex]$900. Therefore, the least amount of additional money Joe can spend is calculated by multiplying the number of blocks by the cost per block:
\[
\text{Additional Cost} = 5 \text{ blocks} \times \$[/tex]900 \text{ per block} = \[tex]$4500
\]
So, the least amount of additional money Joe can spend to get the required rooms is:
\[
\$[/tex]4500
\]
