answer:Let's denote the total number of students in the class as 100% for simplicity. According to the given information: - 75% answered the first question correctly. - 30% answered the second question correctly. - 20% answered neither of the questions correctly. To find the percentage of students who answered both questions correctly, we need to find the intersection of the two groups (those who answered the first question correctly and those who answered the second question correctly). First, let's find out the percentage of students who answered at least one question correctly. Since 20% answered neither, then 100% - 20% = 80% answered at least one question correctly. Now, we can use the principle of inclusion-exclusion to find the percentage who answered both correctly: Percentage who answered both correctly = (Percentage who answered the first question correctly) + (Percentage who answered the second question correctly) - (Percentage who answered at least one question correctly) Plugging in the values we have: Percentage who answered both correctly = 75% + 30% - 80% Percentage who answered both correctly = 105% - 80% Percentage who answered both correctly = 25% So, boxed{25%} of the students answered both questions correctly.

answer:Consider the quadratic equation with respect to ( x ): [ 8x^2 - mx + (m-6) = 0 ] We need to determine the conditions for the values of ( m ) such that the roots of the equation behave in the following ways: 1. Both roots are positive. 2. Both roots are negative. 3. The roots have opposite signs. We will use the properties of quadratic equations and the relationships between the coefficients and the roots. For a quadratic equation of the form ( ax^2 + bx + c = 0 ) with roots ( alpha ) and ( beta ), the following relationships hold: - The sum of the roots is given by ( alpha + beta = -frac{b}{a} ). - The product of the roots is given by ( alphabeta = frac{c}{a} ). # 1. Both Roots are Positive For both roots to be positive, the following conditions must be met: - The sum of the roots must be positive: [ alpha + beta > 0 ] - The product of the roots must be positive: [ alphabeta > 0 ] Using the properties of the coefficients for the given equation ( 8x^2 - mx + (m-6) = 0 ): [ alpha + beta = frac{m}{8} ] [ alphabeta = frac{m-6}{8} ] Hence, we require: [ frac{m}{8} > 0 implies m > 0 ] [ frac{m-6}{8} > 0 implies m - 6 > 0 implies m > 6 ] Additionally, for both roots to be real and distinct, the discriminant must be non-negative: [ Delta = b^2 - 4ac = m^2 - 4 cdot 8 cdot (m-6) = m^2 - 32m + 192 geq 0 ] Solving this inequality: [ m^2 - 32m + 192 geq 0 ] [ (m-16)^2 - 64 geq 0 ] [ (m-16-8)(m-16+8) geq 0 ] [ (m-24)(m-8) geq 0 ] This quadratic inequality holds when ( m leq 8 ) or ( m geq 24 ). However, combining this with the requirement ( m > 6 ), we refine the solution to: [ 6 < m leq 8 quad text{or} quad m geq 24 ] Thus, the range of ( m ) values for both positive roots is: [ 6 < m leq 8 quad text{or} quad m geq 24 ] # 2. Both Roots are Negative For both roots to be negative, the following conditions must be met: - The sum of the roots must be negative: [ alpha + beta < 0 ] - The product of the roots must be positive: [ alphabeta > 0 ] Using the given relationships: [ frac{m}{8} < 0 implies m < 0 ] [ frac{m-6}{8} > 0 implies m > 6 ] There is no overlap between ( m < 0 ) and ( m > 6 ), thus there is no value of ( m ) that satisfies both conditions simultaneously. Hence, there is no value of ( m ) such that both roots are negative: [ m text{ does not exist} ] # 3. Roots Have Opposite Signs For the roots to have opposite signs, the product of the roots must be negative: [ alphabeta < 0 ] Using the relationship: [ frac{m-6}{8} < 0 implies m - 6 < 0 implies m < 6 ] Therefore, the range of ( m ) values for the roots to have opposite signs is: [ m < 6 ] # Conclusion: [ boxed{(1) 6 < m leqslant 8 text{ or } m geqslant 24} ] [ boxed{(2) text{No values of } mtext{ exist}} ] [ boxed{(3) m < 6} ]

answer:To find the distance travelled downstream, we need to calculate the effective speed of the boat when it's moving downstream. The effective speed is the sum of the boat's speed in still water and the rate of the current. Speed of boat in still water = 42 km/hr Rate of current = 5 km/hr Effective speed downstream = Speed of boat in still water + Rate of current Effective speed downstream = 42 km/hr + 5 km/hr Effective speed downstream = 47 km/hr Now, we need to convert the time travelled from minutes to hours to match the units of speed (km/hr). Time travelled downstream = 44 minutes To convert minutes to hours, we divide by 60 (since there are 60 minutes in an hour). Time in hours = Time in minutes / 60 Time in hours = 44 / 60 Time in hours = 0.7333 hours (approximately) Now we can calculate the distance travelled downstream using the formula: Distance = Speed × Time Distance downstream = Effective speed downstream × Time in hours Distance downstream = 47 km/hr × 0.7333 hours Distance downstream ≈ 34.4651 km Therefore, the distance travelled downstream is approximately boxed{34.47} km.

answer:1. **Volume of the cylinder**: Given that the volume ( V ) of the cylinder is ( 72pi ) cm(^3), and the formula for the volume of a cylinder is ( V = pi r^2 h ), we can express the volume as: [ pi r^2 h = 72pi ] Simplifying, we get: [ r^2 h = 72 ] 2. **Volume of the sphere**: The formula for the volume of a sphere is ( V = frac{4}{3} pi r^3 ). Assuming the sphere has the same radius as the cylinder, we need to find ( r ) first. From ( r^2 h = 72 ), choosing an appropriate value for ( h ) to simplify, let's assume ( h = r ) (which would make the cylinder's height equal to its diameter). Then: [ r^3 = 72 quad text{and} quad r = sqrt[3]{72} ] 3. **Substitute ( r ) in the sphere’s volume formula**: [ V = frac{4}{3} pi (sqrt[3]{72})^3 ] Therefore, the volume ( V ) simplifies to: [ V = frac{4}{3} pi times 72 = boxed{96pi} text{ cm}^3 ] Conclusion: The volume of the sphere with the same radius as the cylinder is ( boxed{96pi} ) cm(^3).