desperate need of LOTS of help!
ok i have a few problems for calculus 1. my friend that was supposed to help me is nowhere to be found, so in desperation i've turned to the internet. if you can help me on any of these it would be GREATLY APPRECIATED!
1. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 24 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
Enter your answers correct to two decimal places. Remember to take into account in your calculations that there is only half of a circle. The length of the bottom of the rectangle is __ feet. The length of each of the sides of the rectangle is __ feet.
2. The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, how far in ft from the stronger source should an object be placed on the line between the sources so as to receive the least TOTAL illumination (the sum of the illumination from each source)?
Please enter your answer as a number without the units.
3. Find an equation of the line through the point (8, 16) that cuts off the least area from the first quadrant. Please enter your answer in the slope-intercept form.
4. A painting in an art gallery has height h= 99 cm and is hung so that its lower edge is a distance d=10 cm above the eye of an observer (as seen in the figure below). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle theta subtended at his eye by the painting?) Round the result to the nearest hundredth. [Ok, the drawing is of a right triangle, with the hypotenuse extending from lower left to upper right. If that makes any sense. There is a line from the lower left corner that goes to the right side. The lower portion of the right side is d, and the upper is h. Theta is in the triangle that has side h, if it matters.]
5. A steel pipe is being carried down a hallway 11 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 9 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? Round the answer to the nearest hundredth. [Imagine an L shaped hallway...]
6. Atmospheric pressure V decreases as altitude h increases. At a temperature of the pressure is 101.6kilopascals kPa at sea level, 86.5 kPa at h = 1 km, and 78.7 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Please round your answer to the nearest tenth.
7. The circumference of a sphere was measured to be 92 cm with a possible error of 0.3cm.
Use differentials to estimate the maximum error in the calculated surface area. Please round the answer to the nearest tenth.
What is the relative error in the calculated surface area? Please round the answer to the nearest tenth.
Use differentials to estimate the maximum error in the calculated volume. Please round the answer to the nearest tenth.
What is the relative error in the calculated volume? Please round the answer to the nearest tenth.
1. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 24 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
Enter your answers correct to two decimal places. Remember to take into account in your calculations that there is only half of a circle. The length of the bottom of the rectangle is __ feet. The length of each of the sides of the rectangle is __ feet.
2. The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, how far in ft from the stronger source should an object be placed on the line between the sources so as to receive the least TOTAL illumination (the sum of the illumination from each source)?
Please enter your answer as a number without the units.
3. Find an equation of the line through the point (8, 16) that cuts off the least area from the first quadrant. Please enter your answer in the slope-intercept form.
4. A painting in an art gallery has height h= 99 cm and is hung so that its lower edge is a distance d=10 cm above the eye of an observer (as seen in the figure below). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle theta subtended at his eye by the painting?) Round the result to the nearest hundredth. [Ok, the drawing is of a right triangle, with the hypotenuse extending from lower left to upper right. If that makes any sense. There is a line from the lower left corner that goes to the right side. The lower portion of the right side is d, and the upper is h. Theta is in the triangle that has side h, if it matters.]
5. A steel pipe is being carried down a hallway 11 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 9 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? Round the answer to the nearest hundredth. [Imagine an L shaped hallway...]
6. Atmospheric pressure V decreases as altitude h increases. At a temperature of the pressure is 101.6kilopascals kPa at sea level, 86.5 kPa at h = 1 km, and 78.7 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Please round your answer to the nearest tenth.
7. The circumference of a sphere was measured to be 92 cm with a possible error of 0.3cm.
Use differentials to estimate the maximum error in the calculated surface area. Please round the answer to the nearest tenth.
What is the relative error in the calculated surface area? Please round the answer to the nearest tenth.
Use differentials to estimate the maximum error in the calculated volume. Please round the answer to the nearest tenth.
What is the relative error in the calculated volume? Please round the answer to the nearest tenth.