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At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 $ lb/in^2 $. Below the surface, the water pressure increases by 4.34 $ lb/in^2 $ for every 10 ft of descent.
(a) Express the water pressure as a function of the depth below the ocean surface.(b) At what depth is the pressure 100 $ lb/in^2 $?
a. $434 d+15$b. $196 \mathrm{ft}$
02:18
Jeffrey P.
Calculus 1 / AB
Calculus 2 / BC
Calculus 3
Chapter 1
Functions and Models
Section 2
Mathematical Models: A Catalog of Essential Functions
Functions
Integration Techniques
Partial Derivatives
Functions of Several Variables
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all right in this problem, we're going to find a linear function to represent the relationship between the pressure and the depth underwater in the ocean. And so death is being measured in feet and pressure is being measured in pounds per square inch. And I'm going to use D and P for my variables. And we're told that at the service when the depth is zero, the pressure is £15 per square inch. That would serve as our y intercept. Why intercept always has an ex coordinative zero. And we're also told that for every 10 feet of D sent the water pressure increases by £4.34 per square inch. Since we're going down, we're going to make that a negative. So we have a rate of £4.34 per square inch over negative, 10 10 feet down. And if we simplify that rate, we get negative 0.434 Now that would be our slope, the rate of change. So if we put those two together into the equation of a line using P for R Y and using D for our X, we get P equals negative 0.434 times D plus 15. So there's our function. Okay, now suppose we know that the pressure is £100 per square inch. Then what depth is that? Diver? So that would be part B. So what we're going to do is substitute 100 for the pressure. And so for the depth. So a substitute 100 into the equation. Subtract 15 from both sides and divide both sides by negative 0.434 And if you want to round it to the nearest whole number, you get approximately 196. It's negative 196 feet on the negative indicates that the person is 196 feet below sea level or below surface level.
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