\( 1< \) The equation \( \mathrm{P}=1+\frac{\mathrm{d}}{33} \) gives the pressure, P , in atmospheres (atm) at a depth of d feet in the sea. For what depths d is the pressure at least \( 87.5 \%, 17.5 \) of 20 points

To find the depths \( d \) at which the pressure \( P \) is at least \( 87.5\% \) of 20 points, we first need to calculate \( 87.5\% \) of 20 points. \[ 87.5\% \text{ of } 20 = 0.875 \times 20 = 17.5 \] Next, we set up the inequality based on the equation given for pressure: \[ P = 1 + \frac{d}{33} \] We want to find the values of \( d \) such that: \[ P \geq 17.5 \] Substituting the expression for \( P \) into the inequality gives: \[ 1 + \frac{d}{33} \geq 17.5 \] Now, we can solve for \( d \): 1. Subtract 1 from both sides: \[ \frac{d}{33} \geq 17.5 - 1 \] \[ \frac{d}{33} \geq 16.5 \] 2. Multiply both sides by 33: \[ d \geq 16.5 \times 33 \] Calculating \( 16.5 \times 33 \): \[ d \geq 544.5 \] Thus, the depth \( d \) must be at least \( 544.5 \) feet for the pressure to be at least \( 17.5 \) atm. In conclusion, the depth \( d \) must satisfy: \[ d \geq 544.5 \text{ feet} \]