A sphere with diameter 1 unit is enclosed in a cube of side 1 unit each. Find the unoccupied volume remaining inside the cube.
Show Hint
- \( \frac{1}{4} \)
- \( 2\pi \)
- \( \frac{\pi}{6} - 1 \)
- \( 1 - \frac{\pi}{4} \)
- \( 1 - \frac{\pi}{6} \)
The Correct Option is
Solution and Explanation
The volume \( V_{\text{cube}} \) of the cube with side 1 unit is given by: \[ V_{\text{cube}} = 1^3 = 1 \, \text{unit}^3. \]
Step 2: Calculate the volume of the sphere.
The sphere has a diameter of 1 unit, so its radius \( r = \frac{1}{2} \) unit. The volume \( V_{\text{sphere}} \) of the sphere is given by the formula: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{1}{2}\right)^3 = \frac{4}{3} \pi \times \frac{1}{8} = \frac{\pi}{6}. \]
Step 3: Find the unoccupied volume inside the cube. The unoccupied volume is the volume of the cube minus the volume of the sphere: \[ V_{\text{unoccupied}} = V_{\text{cube}} - V_{\text{sphere}} = 1 - \frac{\pi}{6}. \]
Top Questions on Geometry
Find the perimeter of Isosceles triangle ABC (below) if mAD = 3 and m\(\angle\)BAC = 55 degrees. Round to the nearest hundredth.
What is f(2) for the graph of f(x) below?
According to the graph below, the greatest change in the profit of the Sports Shack occurred between which two consecutive months?
What is the ratio of the area of triangle ABC to the area of square ADFC if CB=(CF)/4?
A square PQRS is enclosed in another square ABCD. Find the ratio of the area of PQRS to the area of ABCD.
Questions Asked in GRE exam
In early-twentieth-century England, it was fashionable to claim that only a completely new style of writing could address a world undergoing unprecedented transformation — just as one literary critic recently claimed that only the new ”aesthetic of exploratory excess” can address a world undergoing . . . well, you know. Yet in early-twentieth-century England, T. S. Eliot, a man fascinated by the ”presence” of the past, wrote the most innovative poetry of his time. The lesson for today’s literary community seems obvious: a reorientation toward tradition would benefit writers no less than readers. But if our writers and critics indeed respect the novel’s rich tradition (as they claim to), then why do they disdain the urge to tell an exciting story?
- GRE - 2024
- Reading Comprehension
In the 1970’s, two debates engaged many scholars of early United States history. One focused on the status of women, primarily White women. Turning on the so-called golden age the ory, which posited that during the eighteenth-century colonial era, American women enjoyed a brief period of high status relative to their English contemporaries and to nineteenth-century American women, this debate pitted scholars who believed women’s lives deteriorated after 1800 against those who thought women’s lives had been no better before 1800. At issue were the causes of women’s subordination: were these causes already in place when the English first settled North America or did they emerge with the rise of nineteenth-century industrial capital ism? The second debate, the so-called origins debate, concerned the emergence of racial slavery in the southern colonies: was slavery the inevitable result of the deep-rooted racial prejudice of early British colonists or did racial prejudice arise only after these planters instituted slave labor?
Although these debates are parallel in some respects, key differences distinguished them. Whereas the debate over women’s status revolved around implicit comparisons of colonial women to their counterparts in the antebellum period (1800-1860), thus inviting comment from scholars of both historical periods, the origins debate was primarily confined to a discussion about slavery in colonial America. Second, in contrast to the newness of the debate over women’s status and its continued currency throughout the early 1980’s, the debate over race and slavery, begun in the 1950’s, had lost some of its urgency with the publication of Morgan’s American Slavery, American Freedom (1975), widely regarded as the last word on the subject. Each debate also assumed a different relationship to the groups whose histories it concerned. In its heyday, the origins debate focused mainly on White attitudes toward Africans rather than on Africans themselves. With few exceptions, such as Wood’s Black Majority (1974) and Mullin’s Flight and Rebellion (1972), which were centrally concerned with enslaved African men, most works pertaining to the origins debate focused on the White architects, mostly male, of racial slavery. In contrast, although women’s historians were interested in the institutions and ide ologies contributing to women’s subordination, they were equally concerned with documenting women’s experiences. As in the origins debate, however, early scholarship on colonial women defined its historical constituency narrowly, women’s historians focusing mainly on affluent White women.
Over time, however, some initial differences between the approaches taken by scholars in the two fields faded. In the 1980’s, historians of race and slavery in colonial America shifted their attention to enslaved people; interest in African American culture grew, thereby bringing en slaved women more prominently into view. Historians of early American women moved in similar directions during the decade and began to consider the effect of racial difference on women’s experience.- GRE - 2024
- Reading Comprehension
The most plausible justification for higher taxes on automobile fuel is that fuel consumption harms the environment and thus adds to the costs of traffic congestion. But the fact that burning fuel creates these ”negative externalities” does not imply that no tax on fuel could ever be too high. Economics is precise about the tax that should, in principle, be levied to deal with negative externalities: the tax on a liter of fuel should be equal to the harm caused by using a liter of fuel. If the tax is more than that, its costs (including the inconvenience to those who would rather have used their cars) will exceed its benefits (including any reduction in congestion and pollution).
- GRE - 2024
- Reading Comprehension
Objectively, of course, the various ecosystems that sustain life on the planet proceed independently of human agency, just as they operated before the hectic ascendancy of Homo sapiens. But it is also true that it is difficult to think of a single such system that has not, for better or worse, been substantially modified by human culture. Nor is this simply the work of the industrial centuries. It has been happening since the days of ancient Mesopotamia. It is coeval with the origins of writing, and has occurred throughout our social existence. And it is this irreversibly modified world, from the polar caps to the equatorial forests, that is all the nature we have.
- GRE - 2024
- Reading Comprehension
Music critics have consistently defined James P. Johnson as a great early jazz pianist, originator of the 1920’s Harlem ”stride” style, and an important blues and jazz composer. In addition, however, Johnson was an innovator in classical music, composing symphonic music that incor porated American, and especially African American, traditions.
Such a blend of musical elements was not entirely new: by 1924 both Milhaud and Gershwin had composed classical works that incorporated elements of jazz. Johnson, a serious musician more experienced than most classical composers with jazz, blues, spirituals, and popular music, was particularly suited to expand Milhaud’s and Gershwin’s experiments. In 1927 he completed his first large-scale work, the blues- and jazz-inspired Yamekraw, which included borrowings from spirituals and Johnson’s own popular songs. Yamekraw, premiered successfully in Carnegie Hall, was a major achievement for Johnson, becoming his most frequently performed extended work. It demonstrated vividly the possibility of assimilating contemporary popular music into the symphonic tradition.- GRE - 2024
- Reading Comprehension