Question 1

Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of rad/s.Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is

A)

<strong>Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of rad/s.Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is </strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

11ea8a3b_b6d6_69f3_9fff_8917a9af45d0_TB6897_11

Question 2

A plane flying horizontally at an altitude of 1 mi and a speed of mi/h passes directly over a radar station.Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

A)

<strong>A plane flying horizontally at an altitude of 1 mi and a speed of mi/h passes directly over a radar station.Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.</strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

11ea8a3b_b6d5_cda5_9fff_c3c83492bd73_TB6897_11

Question 3

If two resistors with resistances   and   are connected in parallel,as in the figure,then the total resistance   measured in ohms ($\Omega$),is given by   If   and   are increasing at rates of   and   respectively,how fast is   changing when   and   ? Round the result to the nearest thousandth.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

Using the formula for the equivalent resistance of two resistors in parallel, we have:

$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$

Taking the derivative of both sides with respect to time, we get:

$-\frac{1}{R^2} \cdot \frac{dR}{dt} = -\frac{1}{R_1^2} \cdot \frac{dR_1}{dt} - \frac{1}{R_2^2} \cdot \frac{dR_2}{dt}$

Substituting the given values, we have:

$-\frac{1}{(11ea8a3b_b6d7_2d58_9fff_ad9e5dd805a6_TB6897_11)^2} \cdot \frac{dR}{dt} = -\frac{1}{(11ea8a3b_b6d7_2d56_9fff_f15604721e2a_TB6897_11)^2} \cdot (11ea8a3b_b6d7_546b_9fff_a59587aa90b7_TB6897_11) - \frac{1}{(11ea8a3b_b6d7_0645_9fff_3f62425a56f5_TB6897_11)^2} \cdot (11ea8a3b_b6d7_546c_9fff_9b58ae43715f_TB6897_11)$

Simplifying, we get:

$\frac{dR}{dt} = \frac{(11ea8a3b_b6d7_2d58_9fff_ad9e5dd805a6_TB6897_11)^2}{(11ea8a3b_b6d7_2d56_9fff_f15604721e2a_TB6897_11) \cdot (11ea8a3b_b6d7_0645_9fff_3f62425a56f5_TB6897_11)} \cdot (11ea8a3b_b6d7_546b_9fff_a59587aa90b7_TB6897_11) - \frac{(11ea8a3b_b6d7_2d58_9fff_ad9e5dd805a6_TB6897_11)^2}{(11ea8a3b_b6d7_2d56_9fff_f15604721e2a_TB6897_11) \cdot (11ea8a3b_b6d7_0645_9fff_3f62425a56f5_TB6897_11)} \cdot (11ea8a3b_b6d7_546c_9fff_9b58ae43715f_TB6897_11)$

Plugging in the given values, we get:

$\frac{dR}{dt} \approx 11ea8a3b_b6d7_7b82_9fff_4727d5c44dba_TB6897_11$

Rounding to the nearest thousandth, we get:

$\frac{dR}{dt} \approx 11ea8a3b_b6d7_7b82_9fff_4727d5c44dba_TB6897_11$

Question 4

A baseball diamond is a square with side 90 ft.A batter hits the ball and runs toward first base with a speed of 32 ft/s.At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

Accepted Answer

The answer of A baseball diamond is a square with...

Question 5

Two carts,A and B,are connected by a rope 39 ft long that passes over a pulley (see the figure below).The point Q is on the floor 12 ft directly beneath and between the carts.Cart A is being pulled away from Q at a speed of 2 ft/s.How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

Accepted Answer

The answer of Two carts,A and B,are connected by a...

Question 6

Gravel is being dumped from a conveyor belt at a rate of ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal.How fast is the height of the pile increasing when the pile is ft high? Round the result to the nearest hundredth.

A)

<strong>Gravel is being dumped from a conveyor belt at a rate of ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal.How fast is the height of the pile increasing when the pile is ft high? Round the result to the nearest hundredth. </strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

The answer of Gravel is being dumped from a conveyor...

Question 7

Two cars start moving from the same point.One travels south at mi/h and the other travels west at mi/h.At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

A)

<strong>Two cars start moving from the same point.One travels south at mi/h and the other travels west at mi/h.At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.</strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

To find the rate of change of distance between two moving objects, we need to use the Pythagorean theorem: $d^2 = x^2 + y^2$, where $d$ is the distance between the two objects, $x$ is the distance traveled by one object, and $y$ is the distance traveled by the other object. To find the rate of change of $d$, we need to differentiate both sides of the equation with respect to time:
$$2d\frac{dd}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$$
We are given that one car is traveling south at 50 mi/h and the other car is traveling west at 40 mi/h.
Therefore, after 2 hours, the distance traveled by the first car is:
$$x = 50	ext{ mi/h} \cdot 2	ext{ h} = 100	ext{ mi}$$
Similarly, the distance traveled by the second car is:
$$y = 40	ext{ mi/h} \cdot 2	ext{ h} = 80	ext{ mi}$$
The initial distance between the two cars is 0, since they started from the same point.
Plugging these values into our equation and simplifying, we get:
$$\frac{dd}{dt} = \frac{x}{d}\frac{dx}{dt} + \frac{y}{d}\frac{dy}{dt} = \frac{100	ext{ mi}}{\sqrt{100^2 + 80^2}	ext{ mi}}\cdot 0	ext{ mi/h} + \frac{80	ext{ mi}}{\sqrt{100^2 + 80^2}	ext{ mi}}\cdot(-40	ext{ mi/h}) \approx -30.77	ext{ mi/h}$$
Rounding to the nearest hundredth, we get the answer as B.

Question 8

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat.If the rope is pulled in at a rate of 1 m/s how fast is the boat approaching the dock when it is 7 m from the dock? Round the result to the nearest hundredth if necessary.

Accepted Answer

The answer of A boat is pulled into a dock...

Question 9

The top of a ladder slides down a vertical wall at a rate of m/s .At the moment when the bottom of the ladder is 3 m from the wall,it slides away from the wall at a rate of 0.2 m/s .How long is the ladder?

A)

<strong>The top of a ladder slides down a vertical wall at a rate of m/s .At the moment when the bottom of the ladder is 3 m from the wall,it slides away from the wall at a rate of 0.2 m/s .How long is the ladder?</strong> A) B) C) D) E) None of these <div style=padding-top: 35px>

B)

C)

D)

E) None of these

Accepted Answer

The answer of The top of a ladder slides down...

Question 10

If a snowball melts so that its surface area decreases at a rate of   , find the rate at which the diameter decreases when the diameter is 35 cm.&#10;

Accepted Answer

The answer of If a snowball melts so that its...

Deck 2: Extension G: Limits and Derivatives