Question

a scuba diver wants to dive deeper than $2\frac{1}{2}$ times his previous dive. that means he has to dive at least 80 feet below the surface of the water. this dive is represented by the inequality $2\frac{1}{2}p < -80$, where p is the end location of her previous dive relative to the surface of the water. select from the drop-down to correctly interpret the solution of this inequality. the scuba diver’s previous dive was choose... feet below the surface of the water. choose... at most 32 deeper than 32 less than 32

Explanation:

Step1: Convert mixed number to improper fraction

The mixed number \(2\frac{1}{2}\) can be converted to an improper fraction. \(2\frac{1}{2}=\frac{2\times2 + 1}{2}=\frac{5}{2}\). So the inequality becomes \(\frac{5}{2}p<-80\).

Step2: Solve the inequality for \(p\)

To solve for \(p\), we multiply both sides of the inequality by \(\frac{2}{5}\) (the reciprocal of \(\frac{5}{2}\)). When we multiply both sides of an inequality by a positive number, the direction of the inequality sign remains the same. So we have \(p<-80\times\frac{2}{5}\). Calculating \(-80\times\frac{2}{5}\), we can simplify: \(-80\div5 = - 16\), and \(-16\times2=-32\). So \(p < - 32\). Since \(p\) represents the end location of the previous dive relative to the surface (negative values mean below the surface), \(p < - 32\) means the previous dive was deeper than 32 feet below the surface? Wait, no. Wait, if \(p\) is the end location of the previous dive relative to the surface, and we have \(p < - 32\), that means the previous dive was more than 32 feet below the surface? Wait, no, let's re - examine. Wait, the inequality is \(2\frac{1}{2}p<-80\), where \(p\) is the previous dive's end location. If we solve \(p < - 32\), that means \(p\) (the previous dive's depth below the surface, since it's negative) is less than - 32, which means the depth is greater than 32 feet below the surface? Wait, no, let's think in terms of depth. If the surface is 0, and below the surface is negative. So if \(p=-30\), that's 30 feet below. If \(p = - 33\), that's 33 feet below. So \(p < - 32\) means that the previous dive was deeper than 32 feet below the surface? Wait, no, wait. Wait, the current dive is \(2\frac{1}{2}\) times the previous. The current dive has to be at least 80 feet below, so \(2\frac{1}{2}p<-80\) (since below the surface is negative). Solving for \(p\), we get \(p < - 32\). So \(p\) is the previous dive's depth. So if \(p\) is less than - 32, that means the previous dive was deeper than 32 feet below the surface? Wait, no, - 33 is less than - 32, and - 33 is 33 feet below, which is deeper than 32 feet below. So the correct interpretation is that the previous dive was deeper than 32 feet below the surface? Wait, but let's check the options. The options are "at most 32", "deeper than 32", "less than 32". Wait, maybe I made a mistake in the sign. Wait, maybe the inequality should be \(2\frac{1}{2}p\leq - 80\) (since "at least 80 feet below" would be less than or equal to - 80). But the problem says \(2\frac{1}{2}p < - 80\). Let's re - solve:

\(2\frac{1}{2}p<-80\)

\(\frac{5}{2}p<-80\)

Multiply both sides by \(\frac{2}{5}\):

\(p<-80\times\frac{2}{5}\)

\(p < - 32\)

Since \(p\) is the depth of the previous dive (negative value, as it's below the surface), \(p < - 32\) means that the previous dive's depth (in terms of how far below the surface) is greater than 32 feet. Because if \(p=-33\), it's 33 feet below, which is deeper than 32 feet below. If \(p=-30\), it's 30 feet below, which is less than 32 feet below. So \(p < - 32\) means that the previous dive was deeper than 32 feet below the surface? Wait, no, wait. Wait, the value of \(p\) is negative. So \(p < - 32\) means that \(p\) is a number that is more negative than - 32. So, for example, \(p=-33\) is less than - 32, and - 33 is 33 feet below the surface, which is deeper than 32 feet below the surface. So the correct option is "deeper than 32". Wait, but let's check the inequality again. The problem says "dive deeper than \(2\frac{1}{2}\) times his previous dive. That means he has to dive at least 80 feet below the surface". So the current dive depth is \(2\frac{1}{2}p\) (where \(p\) is previous), and current dive depth is at least 80 feet below, so \(2\frac{1}{2}p\leq - 80\) (since below is negative). But the inequality given is \(2\frac{1}{2}p < - 80\). Anyway, when we solve \(p < - 32\), since \(p\) is the previous dive's depth (negative), \(p < - 32\) implies that the previous dive was deeper than 32 feet below the surface.

Answer:

deeper than 32