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Basic Mathematics Homework HelpQ1: Point M is the midpoint of PQ. If PM = 23x + 5 and MQ = 25x - 4, find the length of PQ.
See AnswerThe length of PQ is twice the length of PM or MQ. Therefore, PQ = 2 * |PM| = 2 * |23x + 5| = 46x + 10.
Q2: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x², y = 1 about y = 6.
See AnswerUsing the method of disks, the volume V = π∫[a,b] [(radius_outer)² - (radius_inner)²] dx. For this problem, V = π ∫[0,1] [(6 - x²)² - (6 - 1)²] dx. The detailed evaluation involves integration techniques and results in the final answer.
Q3: A set designer for a play makes a whale tail out of a semicircle with an isosceles triangle cut out of it. What is the area of the whale tail? [5 marks]
See AnswerThe area of the whale tail is calculated by subtracting the area of the isosceles triangle from the area of the semicircle. Detailed calculations depend on specific dimensions given in the problem.
Q4: A decorator is wallpapering a wall with a circular window of diameter 1.00 m. What is the area of the wall in square feet? (1 m = 3.2808 feet) [6 marks]
See AnswerThe area of the wall minus the circular window is calculated using the area formulas. The circular window's area is π(0.5 m)² converted to square feet. Final answer depends on the size of the wall provided.
Q5: A furniture designer builds a trapezoidal desk with a semicircular cutout. What is the area of the desk? [7 marks]
See AnswerThe area of the desk is the area of the trapezoid minus the area of the semicircular cutout. Detailed calculation involves specific dimensions of the trapezoid and semicircle.
Q6: Find the centroid of the region bounded by y = x² and y = 4. [8 marks]
See AnswerThe centroid is found using the formulas for the center of mass for a plane region: (x̄, ȳ) = (1/Area) ∫∫ (x dA, y dA). The detailed calculation involves integrating within the given bounds.
Q7: Solve the differential equation dy/dx = y² - y. [6 marks]
See AnswerThe solution involves separating variables and integrating both sides. Final answer depends on initial conditions provided.
Q8: Determine the area of this composite figure to the nearest square feet and square metres. [8 marks]
See AnswerThe area is calculated by dividing the composite figure into simpler shapes, finding their areas, and adding them together. Unit conversion from square meters to square feet is done afterwards.
Q9: Solve the integral ∫ (2x³ - x² + 3x - 1) dx. [5 marks]
See AnswerThe integral of the polynomial is calculated as follows: (1/2) * 2x⁴ - (1/3) * x³ + (3/2) * x² - x + C.
Q10: Find the length of the curve y = ln(x) from x = 1 to x = 2. [7 marks]
See AnswerThe length of the curve is found using the formula L = ∫[a,b] sqrt(1 + (dy/dx)²) dx. The detailed calculation results in a final value after integration.
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