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If the diagonal distance across the plot is $\sqrt{450}$ feet, what is the perimeter of the plot, in feet?
The diagonal of a rectangle forms a right triangle with the length and width as its legs. We can use the Pythagorean theorem: $w^2 + (2w)^2 = (\sqrt{450})^2$.
Simplify the equation: $w^2 + 4w^2 = 450$. This combines to $5w^2 = 450$.
Divide both sides by 5: $w^2 = 90$.
Solve for $w$ by taking the square root of both sides: $w = \sqrt{90}$. To simplify $\sqrt{90}$, find the largest perfect square factor of 90, which is 9. So, $w = \sqrt{9 \cdot 10} = 3\sqrt{10}$ feet.
The length is $2w = 2(3\sqrt{10}) = 6\sqrt{10}$ feet.
The perimeter of the rectangle is given by the formula $P = 2(length + width)$.
Substitute the values: $P = 2(6\sqrt{10} + 3\sqrt{10})$.
Combine the like terms inside the parentheses: $P = 2(9\sqrt{10})$.
Multiply to find the perimeter: $P = 18\sqrt{10}$ feet.
The writer wants to add a sentence to this paragraph that best emphasizes how independent film challenges traditional cinematic expectations and provides fresh perspectives. Which of the following sentences, if added at this point, would best accomplish this goal?
Distractor B discusses distribution methods, which, while true about independent film, does not directly relate to how these films challenge traditional perspectives or provide fresh insights into social issues. Distractor C focuses on the subjective reception of films, which is irrelevant to the paragraph's focus on the *content* and *impact* of independent cinema. Distractor D mentions the diversity of genres, which is too general and does not specifically address the paragraph's emphasis on challenging perceptions or offering unique social commentary.
Scientist 1's Viewpoint: Lake Solstice is primarily a result of recent intense volcanic activity. The lake basin formed from a collapsed caldera, and the unusual water chemistry (high dissolved sulfur compounds) and localized geothermal vents are direct evidence of this recent volcanism influencing its ecosystem and depth.
Scientist 2's Viewpoint: Lake Solstice is a classic glacial lake, carved by massive ice sheets during the last ice age. Its extreme depth is due to glacial scour, and the unique aquatic species are adapted to cold, oxygen-rich waters typical of deep glacial lakes. The presence of sulfur compounds and slight geothermal activity are minor, localized phenomena, not the primary drivers of the lake's overall characteristics.
Observed Data:
Distractor 1 is incorrect because while it cites specific data points (elevated temperatures and sulfur compounds), it overstates their significance by claiming they 'definitively confirm volcanic influence as the primary shaper.' The overwhelming evidence from topography, basin shape, and sediment composition points away from primary volcanic formation. Distractor 2 is incorrect because the evidence is not equally balanced; the glacial indicators are far more numerous and widespread than the limited and localized volcanic indicators. Distractor 3 is incorrect because, while there might be conflicting *details*, the question asks which viewpoint is *most strongly supported*. The data provides clear and predominant support for Scientist 2's hypothesis regarding the lake's primary formation.
"The measure of angle BAD are 35 degrees, due to the fact that the sum of the angles in a triangle equals 180 degrees, and the exterior angle theorem implies a relationship."
Which of the following is the best revision of the underlined portion?
Angle ADC is supplementary to angle ADB, meaning angle ADC = 180 - 105 = 75 degrees. In triangle ADC, the sum of angles is 180 degrees, so angle DAC = 180 - (angle C + angle ADC) = 180 - (35 + 75) = 70 degrees. Since segment AD bisects angle BAC, angle BAD must also be 70 degrees, thus confirming the choice that accurately states "is 70 degrees, because angle ADC is supplementary to angle ADB, making angle ADC equal to 75 degrees. Then, in triangle ADC, angle DAC equals 180 - (75 + 35), which is 70 degrees, so angle BAD is also 70 degrees since AD bisects angle BAC."
The option suggesting angle BAD "are 35 degrees, considering that AD bisects angle BAC, so triangle ABC is isosceles with base BC, making angle B equal to angle C" is incorrect because angle bisection alone does not make triangle ABC isosceles or imply angle B equals angle C. The reasoning that angle BAD "is 105 degrees, as angle ADB is 105 degrees and AD bisects angle BAC, which means angle BAD must be equal to angle ADB" lacks any supporting geometric principle that would equate these two angles. Lastly, the choice claiming angle BAD "is 40 degrees, since the sum of angles in triangle ABD must be 180 degrees, and if angle B is 35 degrees, then angle BAD equals 180 - (35 + 105)" makes an erroneous assumption that angle B is 35 degrees, when the problem explicitly states that angle C is 35 degrees.
In this context, the term "allowable production range" refers to which of the following on the coordinate plane?
The "allowable production range" for a system of inequalities refers to the set of all possible solutions that satisfy every condition simultaneously. Therefore, the correct explanation is that it is the region on the coordinate plane containing all points (x, y) that satisfy every given inequality, often called the feasible region. The option stating "only the boundary lines of the inequalities, excluding any interior points" is incorrect because the solution region includes the shaded area within the boundaries, representing all points that satisfy the inequalities. Similarly, the option focusing on "only the points where the boundary lines of the inequalities intersect each other or the axes" is insufficient, as these are merely the vertices of the feasible region, not the entire range. Lastly, the description "the area of the coordinate plane that does not satisfy any of the given conditions" describes the excluded region, not the allowable production range.