Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
The Question Ronan is building an extension to his house to facilitate a social space and games room. One of his passions is watching sport on a big screen, so he has purchased a wide-lens home cinema projector that he can connect to his laptop, instead of buying a large-screen TV. Having read the manufacturer's guide extensively, and researched his options on the internet for optimising picture quality, he decides that he would like to have a viewing area equivalent to having a 120-inch TV mounted on the wall. This means that the width of the viewing area will be 108 inches. The projector has a wide lens that covers an angle of 72°. He needs to position the projector centrally with respect to the viewing area, on a stand that you may assume is at the correct height for optimum picture quality. Ronan sketches the situation in Figure 7, where all lengths are measured in inches. The point P represents the projector's position on the stand. Points A and B are the furthest reaches of the viewing area (the width). The point C is the centre point of the screen's width, directly in front of the projector. The length of PA is equal to the length of PB. Angle APB is 72°, so angle APC is 36°. 108 B 36 P. Figure 7 Assuming that all aspects of picture quality are satisfactory, how far from centre point C should the projector be placed in order to fill the full screen width of 108 inches?

The Question Ronan is building an extension to his house to facilitate a social space and games room. One of his passions is watching sport on a big screen, so he has purchased a wide-lens home cinema projector that he can connect to his laptop, instead of buying a large-screen TV. Having read the manufacturer's guide extensively, and researched his options on the internet for optimising picture quality, he decides that he would like to have a viewing area equivalent to having a 120-inch TV mounted on the wall. This means that the width of the viewing area will be 108 inches. The projector has a wide lens that covers an angle of 72°. He needs to position the projector centrally with respect to the viewing area, on a stand that you may assume is at the correct height for optimum picture quality. Ronan sketches the situation in Figure 7, where all lengths are measured in inches. The point P represents the projector's position on the stand. Points A and B are the furthest reaches of the viewing area (the width). The point C is the centre point of the screen's width, directly in front of the projector. The length of PA is equal to the length of PB. Angle APB is 72°, so angle APC is 36°. 108 B 36 P. Figure 7 Assuming that all aspects of picture quality are satisfactory, how far from centre point C should the projector be placed in order to fill the full screen width of 108 inches?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Do you think that the student’s approach to solving the problem is the
most efficient method? Give a reason for your answer.

The Question Ronan is building an extension to his house to facilitate a social space and games room. One of his passions is watching sport on a big screen, so he has purchased a wide-lens home cinema projector that he can connect to his laptop, instead of buying a large-screen TV. Having read the manufacturer's guide extensively, and researched his options on the internet for optimising picture quality, he decides that he would like to have a viewing area equivalent to having a 120-inch TV mounted on the wall. This means that the width of the viewing area will be 108 inches. The projector has a wide lens that covers an angle of 72°. He needs to position the projector centrally with respect to the viewing area, on a stand that you may assume is at the correct height for optimum picture quality. Ronan sketches the situation in Figure 7, where all lengths are measured in inches. The point P represents the projector's position on the stand. Points A and B are the furthest reaches of the viewing area (the width). The point C is the centre point of the screen's width, directly in front of the projector. The length of PA is equal to the length of PB. Angle APB is 72°, so angle APC is 36°. 108 B 36 Figure 7 Assuming that all aspects of picture quality are satisfactory, how far from centre point C should the projector be placed in order to fill the full screen width of 108 inches?
Transcribed Image Text:The Question Ronan is building an extension to his house to facilitate a social space and games room. One of his passions is watching sport on a big screen, so he has purchased a wide-lens home cinema projector that he can connect to his laptop, instead of buying a large-screen TV. Having read the manufacturer's guide extensively, and researched his options on the internet for optimising picture quality, he decides that he would like to have a viewing area equivalent to having a 120-inch TV mounted on the wall. This means that the width of the viewing area will be 108 inches. The projector has a wide lens that covers an angle of 72°. He needs to position the projector centrally with respect to the viewing area, on a stand that you may assume is at the correct height for optimum picture quality. Ronan sketches the situation in Figure 7, where all lengths are measured in inches. The point P represents the projector's position on the stand. Points A and B are the furthest reaches of the viewing area (the width). The point C is the centre point of the screen's width, directly in front of the projector. The length of PA is equal to the length of PB. Angle APB is 72°, so angle APC is 36°. 108 B 36 Figure 7 Assuming that all aspects of picture quality are satisfactory, how far from centre point C should the projector be placed in order to fill the full screen width of 108 inches?
The student's incorrect attempt We first calculate the angle at A, using the fact that all three angles in a triangle add up to 180°. We have A = 180 - (90 +36) =54, so the angle at A is 54°, Using the Sine Rule in triangle ABP, we have a sinA sin? SO a = x sinA sinP I08 x sin 54° %3D sin 720 = 91.81.... So the length of BP is 92 inches (to the nearest inch). The length of BP is the same as the length Of AP, and trianale ACP is right-angled, so we have adj_ CP cos 36° hyp 92' SO 92 CP = = 13.1... Cos 36° Therefore to the nearest inch, the projector should Be placed t inches away from the centre of the viewing area.
Transcribed Image Text:The student's incorrect attempt We first calculate the angle at A, using the fact that all three angles in a triangle add up to 180°. We have A = 180 - (90 +36) =54, so the angle at A is 54°, Using the Sine Rule in triangle ABP, we have a sinA sin? SO a = x sinA sinP I08 x sin 54° %3D sin 720 = 91.81.... So the length of BP is 92 inches (to the nearest inch). The length of BP is the same as the length Of AP, and trianale ACP is right-angled, so we have adj_ CP cos 36° hyp 92' SO 92 CP = = 13.1... Cos 36° Therefore to the nearest inch, the projector should Be placed t inches away from the centre of the viewing area.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS