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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.4.73b
Chapter 6, Problem 6.4.73b

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.


b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

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First, visualize the problem by drawing a right circular cone with radius 1 and height 3, and then inscribe a cube inside it so that the cube touches the cone's sides and base.
Set up a coordinate system with the cone's vertex at the origin and the base at height 3. Express the radius of the cone's cross-section at any height \( y \) as a linear function: \( r(y) = 1 - \frac{y}{3} \).
Let the side length of the cube be \( s \). Position the cube so that its base lies on the cone's base (at \( y = 3 \)) and its top is at \( y = 3 - s \). The half-width of the cube at height \( y \) must be less than or equal to the radius of the cone at that height.
Use the relationship between the cube's half side length \( \frac{s}{2} \) and the cone's radius at the cube's top \( r(3 - s) \) to set up the equation \( \frac{s}{2} = r(3 - s) = 1 - \frac{3 - s}{3} \).
Solve this equation for \( s \) to find the side length of the cube, then compute the volume of the cube using \( V = s^3 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inscribed Solids and Geometric Constraints

Understanding how a solid, like a cube, fits inside another shape, such as a cone, requires analyzing the geometric constraints. This involves relating dimensions of the inscribed solid to the dimensions of the outer shape, ensuring all vertices lie within or on the boundary of the cone.
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Volume Formulas and Optimization

Calculating the volume of the cube involves using the formula V = s³, where s is the side length. To find the largest possible cube inside the cone, one must optimize s under the given constraints, often requiring setting up equations that relate s to the cone's radius and height.
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Using Calculus for Maximization Problems

Calculus techniques, such as derivatives, help find maximum or minimum values of functions. In this problem, calculus can be used to maximize the cube's volume by differentiating the volume function with respect to the cube's side length and solving for critical points.
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Related Practice
Textbook Question

Power and energy The terms power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶ J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1W=1 J/s). Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6×10⁶ J. Suppose the power function of a large city over a 24-hr period is given by P(t) = E'(t) = 300 - 200 sin πt/12, where P is measured in megawatts and t=0 corresponds to 6:00 P.M. (see figure).


b. Burning 1 kg of coal produces about 450 kWh of energy. How many kilograms of coal are required to meet the energy needs of the city for 1 day? For 1 year? 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. If a region is revolved about the y-axis, then the shell method must be used.

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Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

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Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

b. If the length is doubled, is the required work doubled? Explain.

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Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [0,3]?

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Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

c. If the radius is doubled, is the required work doubled? Explain.

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