We need to prove that, for r < s , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where C is the point on the parabola at which the tangent line is parallel to the secant line A E ¯ . (a) We need to show that C has x coordinate ( r + s ) / 2 . (b) We need to prove that ABDE has area ( s − r ) 3 4 by viewing it as a parallelogram of height s − r and base of length C F ¯ . (c) We need to show that the triangle ACE has area ( s − r ) 3 8 by observing it has the same height and base as the parallelogram. (d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Answer 1

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Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Answer 2

Question

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Answer 3

Question

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Answer 4

Question

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Answer 5

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

Answer 6

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

Answer 7

Chapter 5.4, Problem 51E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

Solution Summary: The author proves that for r s, the area of the shaded region in Figure 9 is equal to the four-thirds of triangle ACE.

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