Question

Two straight roads intersect at right angles. A car is traveling south toward the intersection while a radar-monitored-speed sign is positioned 1/4 mile east of the intersection. The speed sign provides both the distance from the sign to an approaching car and the rate at which the distance between the sign and the car is changing. At a certain time, the speed sign shows that the approaching car is exactly 1/2 mile away from the sign and that the rate at which the distance between the car and the sign is decreasing is 60 miles per hour. Which of the following is true about the distance between the car and the intersection at this instant? A The distance is increasing at a rate of 120/?3 miles per hour. B The distance is decreasing at a rate of 120/?3 miles per hour. C The distance is increasing at a rate of 30/?3 miles per hour. D The distance is decreasing at a rate of 30/?3 miles per hour.

          Two straight roads intersect at right angles. A car is traveling south toward the intersection while a radar-monitored-speed sign is positioned 1/4 mile east of the intersection. The speed sign provides both the distance from the sign to an approaching car and the rate at which the distance between the sign and the car is changing. At a certain time, the speed sign shows that the approaching car is exactly 1/2 mile away from the sign and that the rate at which the distance between the car and the sign is decreasing is 60 miles per hour. Which of the following is true about the distance between the car and the intersection at this instant?
A The distance is increasing at a rate of 120/?3 miles per hour.
B The distance is decreasing at a rate of 120/?3 miles per hour.
C The distance is increasing at a rate of 30/?3 miles per hour.
D The distance is decreasing at a rate of 30/?3 miles per hour.

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Two straight roads intersect at right angles. A car is traveling south toward the intersection while a radar-monitored-speed sign is positioned 1/4 mile east of the intersection. The speed sign provides both the distance from the sign to an approaching car and the rate at which the distance between the sign and the car is changing. At a certain time, the speed sign shows that the approaching car is exactly 1/2 mile away from the sign and that the rate at which the distance between the car and the sign is decreasing is 60 miles per hour. Which of the following is true about the distance between the car and the intersection at this instant?
A The distance is increasing at a rate of 120/?3 miles per hour.
B The distance is decreasing at a rate of 120/?3 miles per hour.
C The distance is increasing at a rate of 30/?3 miles per hour.
D The distance is decreasing at a rate of 30/?3 miles per hour.

Added by Daniel T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Keondre Parker

03/03/2023

Step 1

Let x be the distance between the car and the intersection, and y be the distance between the car and the speed sign. We are given that y = 1 mile. Show more…

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Two straight roads intersect at right angles. A car is traveling south toward the intersection while a radar-monitored speed sign is positioned 1/4 mile east of the intersection. The speed sign provides both the distance from the sign to an approaching car and the rate at which the distance between the sign and the car is changing. At a certain time, the speed sign shows that the approaching car is exactly 1/2 mile away from the sign and that the rate at which the distance between the car and the sign is decreasing is 60 miles per hour. Which of the following is true about the distance between the car and the intersection at this instant? The distance is increasing at a rate of 120/√3 miles per hour. The distance is decreasing at a rate of 120/√3 miles per hour. The distance is increasing at a rate of 30/√3 miles per hour. The distance is decreasing at a rate of 30/√3 miles per hour.

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Transcript

00:01 Alright, so let's start off by drawing an image.

00:04 So we have a car traveling south towards the intersection.

00:11 And then we have some sign, radar sign here.

00:16 So we know that the radar sign is one -fourth of a mile east of the intersection.

00:29 And then we know that when the car is exactly half a mile away from the sign, so in this case, half a mile away from the sign.

00:40 A mile then the right here is negative which is decreasing at 60 miles per hour so the rate is negative negative 60 miles per hour so in this case we want to figure out what's true about the distance between a car and the intersection at this instance so in this case why so this is set up like a the thagorean theorem.

01:15 So we're going to have one -fourth squared plus y -squared is equal to one -half squared.

01:33 So in this case, when we simplify this, this is going to give us one -sixteenth plus y -squared equal to one -fourth, or we can write this as four over sixteen.

01:52 So we can have y squared is equal to three over 16 or y is equal to the square root of three over 16 so we want to to know which is true about the following distance between a car and intersection at this moment so we need to find the derivative of our pythagorean theorem so we have two x dx plus two y d y plus two r d r so in this case we know i would need to plug in some bits of information so we have two times the x distance which is one -fourth but the change in x is zero so you're have to worry about that section you have two y which we found y to be we regret this as the square root of three over four so square root of three over four and then we're looking for to change the distance for the car intersection and then two and r we know is one half and then the change in r is negative 60 so let's simplify some components we have two and four turns it to and two, that just turns it to zero.

03:46 So we have square root of three over two, d .y...

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