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A rectangular solid above has dimensions 3, a, and b, where a and b are integers. Which of the following CANNOT be the areas of three different faces of this solid?
  • a)
    15, 18, and 30
  • b)
    18, 24, and 48
  • c)
    12, 15, and 24
  • d)
    15, 24, and 40
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A rectangular solid above has dimensions 3, a, and b, where a and b ar...
On the drawing, we should first mark the areas of the three faces. The front and back faces both have an area of 3a. The left and right faces both have an area of 3b. The top and bottom faces both have an area of ab. We should now try to find integer values for a and b so that these areas match those given in the choices.
(A) 15, 18, and 30 This is possible if a = 5 and b = 6.
(B) 18, 24, and 48 This is possible if a = 6 and b = 8.
(C) 12, 15, and 24 This cannot work for any integer
values of a and b.
(D) 15, 24, and 40 This is possible if a = 5 and b = 8.
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Most Upvoted Answer
A rectangular solid above has dimensions 3, a, and b, where a and b ar...
On the drawing, we should first mark the areas of the three faces. The front and back faces both have an area of 3a. The left and right faces both have an area of 3b. The top and bottom faces both have an area of ab. We should now try to find integer values for a and b so that these areas match those given in the choices.
(A) 15, 18, and 30 This is possible if a = 5 and b = 6.
(B) 18, 24, and 48 This is possible if a = 6 and b = 8.
(C) 12, 15, and 24 This cannot work for any integer
values of a and b.
(D) 15, 24, and 40 This is possible if a = 5 and b = 8.
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Question is based on the following passage.This passage is adapted from Sabrina Richards, “Pleasant tothe Touch.” ©2012 by The Scientist.In the early 1990s, textbooks acknowledged thathumans had slow-conducting nerves, but assertedthat those nerves only responded to two types ofLine stimuli: pain and temperature. Sensations of pressure5 and vibration were believed to travel only alongmyelinated, fast-signaling nerve fibers, which alsogive information about location. Experimentsblocking nerve fibers supported this notion.Preventing fast fibers from firing (either by clamping10 the relevant nerve or by injecting the local anestheticlidocain e) seemed to eliminate the sensation ofpressure altogether, but blocking slow fibers onlyseemed to reduce sensitivity to warmth or a smallpainful shock.15 Håkan Olausson and his Gothenburg Universitycolleagues Åke Vallbo and Johan Wessbergwondered if slow fibers responsive to gentle pressuremight be active in humans as well as in othermammals. In 1993, they corralled 28 young20 volunteers and recorded nerve signals while gentlybrushing the subjects’ arms with their fingertips.Using a technique called microneurography, inwhich a fine filament is inserted into a single nerve tocapture its electrical impulses, the scientists were able25 to measure how quickly—or slowly—the nervesfired. They showed that soft stroking promptedtwo different signals, one immediate and onedelayed. The delay, Olausson explains, means thatthe signal from a gentle touch on the forearm will30 reach the brain about a half second later. This delayidentified nerve impulses traveling at speedscharacteristic of slow, unmyelinated fibers—about1 meter/second—confirming the presence of thesefibers in human hairy skin. (In contrast, fast-35 conducting fibers, already known to respond totouch, signal at a rate between 35 and 75 m/s.)Then, in 1999, the group looked more closely atthe characteristics of the slow fibers. They namedthese “low-threshold” nerves “C-tactile,” or CT,40 fibers, said Olausson, because of their “exquisitesensitivity” to slow, gentle tactile stimulation, butunresponsiveness to noxious stimuli like pinpricks.But why exactly humans might have such fibers,which respond only to a narrow range of rather45 subtle stimuli, was initially mystifying. Unlike othertypes of sensory nerves, CT fibers could be foundonly in hairy human skin—such as the forearm andthigh. No amount of gentle stroking of hairless skin,such as the palms and soles of the feet, prompted50 similar activity signatures. Olausson and hiscolleagues decided that these fibers must beconveying a different dimension of sensoryinformation than fast-conducting fibers.Although microneurography can give55 information about how a single nerve responds togentle brushing and pressure, it cannot tease outwhat aspect of sensation that fiber relays, saysOlausson. He wanted to know if that same slownerve can distinguish where the brush touches the60 arm, and whether it can discern the differencebetween a goat-hair brush and a feather. Mostimportantly, could that same fiber convey a pleasantsensation?To address the question, Olausson’s group sought65 out a patient known as G.L. who had an unusualnerve defect. More than 2 decades earlier, she haddeveloped numbness across many parts of her bodyafter taking penicillin to treat a cough and fever.Testing showed that she had lost responsiveness to70 pressure, and a nerve biopsy confirmed that G.L.’squick-conducting fibers were gone, resulting in aninability to sense any pokes, prods, or pinpricksbelow her nose. But she could still sense warmth,suggesting that her slow-conducting unmyelinated75 fibers were intact.Upon recruiting G.L., Olausson tested her bybrushing her arm gently at the speed of between2–10 centimeters per second. She had more troubledistinguishing the direction or pressure of the brush80 strokes than most subjects, but reported feeling apleasant sensation. When the researchers triedbrushing her palm, where CT fibers are not found,she felt nothing.Olausson used functional MRI studies to examine85 which areas of the brain lit up when G.L.’s arm wasgently brushed to activate CT fibers. In normalsubjects, both the somatosensory and insular corticeswere activated, but only the insular cortex [whichprocesses emotion] was active when researchers90 brushed G.L.’s arm. This solidified the notion thatCT fibers convey a more emotional quality of touch,rather than the conscious aspect that helps usdescribe what we are sensing. CT fibers, it seemed,specifically provide pleasurable sensations.Q.According to the passage, G.L. differed from Olausson’s other test subjects in terms of the

Question refer to the following information.Performance Banner Company creates promotional banners that include company logos. The Zypz Running Shoe Company would like a 4-foot high and 20-foot long rectangular banner that includes its logo, which has a height-to-length ratio of 5 : 8.Performance Banner Company charges its customers $1.20 per square foot for the banner material, $2.50 per square foot of any printed logo, and $32 in fixed costs per banner. The Zypz Running Shoe Company is considering two options for the banner: one with a single logo, and another with two logos. If these logos are all to be the same size as described in Part 1, what percent of the banner costs would the company save by choosing the single-logo option instead of the two-logo option? (Ignore the % symbol when entering into the grid. For example, enter 27% as 27.) Correct answer is '25'. Can you explain this answer?

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A rectangular solid above has dimensions 3, a, and b, where a and b are integers. Which of the following CANNOT be the areas of three different faces of this solid?a)15, 18, and 30b)18, 24, and 48c)12, 15, and 24d)15, 24, and 40Correct answer is option 'C'. Can you explain this answer?
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