Math AI Solver
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Expert help when you need it- Q1: Evaluate ∫cos^3(x) sin(x) dxAnswerTo evaluate the integral ∫cos^3(x) sin(x) dx, we can use a substitution method. ### Step 1: Choose a substitution Let u = cos(x). Then, the derivative is du = -sin(x)dx or -du = sin(x)dx. ### Step 2: Substitute in the integral Rewriting the integral in terms of u : ∫cos^3(x) sin(x) dx = ∫ u^3 (-du) = - ∫ u^3 du. ### Step 3: Integrate Now, integrate - ∫ u^3 du : - ∫u^3 du = -u^4/4 + C = -cos^4(x)/4 + C. ### Step 4: Final result Thus, the final result of the integral is: ∫cos^3(x) sin(x) dx = -cos^4(x)/4 + C.
- Q2: solve the system of equations: ( x + y + z = 25 ) , ( 5x + 3y + 2z = 0 ),( y - z = 6 )AnswerTo solve the system of equations: 1. ( x + y + z = 25 ) 2. ( 5x + 3y + 2z = 0 ) 3. ( y - z = 6 ) First, from Equation 3, express y in terms of ( z ): y = z + 6 Substituting into Equations 1 and 2: **From Equation 1:** x + (z + 6) + z = 25 implies x + 2z = 19 (1) **From Equation 2:** 5x + 3(z + 6) + 2z = 0 implies 5x + 5z + 18 = 0 implies x + z = -18/5 (2) Now, subtract Equation (2) from Equation (1): (x + 2z) - (x + z) = 19 + 18/5 z = 19 + 18/5 =113/5 Substituting z back to find y : y = z + 6 = 143/5 And substituting z into Equation (2) to find x : x = -131/5 Final results: x = -131/5 y = 143/5 z = 113/5
- Q3:A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2,727 typical miles?AnswerFirst, calculate the total number of gallons of gas needed for the trip by dividing the total miles by the car's average miles per gallon. Total gallons = 2727 miles /27 miles per gallon Next, calculate the total cost of the gas by multiplying the total gallons by the cost per gallon. Total cost=Gallons needed×Cost per gallon =101 gallons×4.04 dollars per gallon ≈408.04 dollars Thus, the cost of gas for the car to travel 2,727 miles is approximately $408.04.
- Q4:64 men can complete a piece of work in 40 days. In how many days can 32 men complete the same piece of work?AnswerTo determine how many days it will take for 32 men to complete the same piece of work, we can use the concept of man-days, which refers to the total amount of work done in terms of the number of men and the number of days. First, calculate the total man-days required to complete the work: Total man-days = Number of men*Number of days = 64 men * 40 days = 2560 man-days Next, we need to find out how many days it will take for 32 men to complete the same amount of work: Let d be the number of days for 32 men to complete the work. The equation can be set up as follows: 32 men *d days = 2560 man-days Now, solve for d : d = 2560 man-days/32 men = 80 days Therefore, it will take **80 days** for 32 men to complete the same piece of work.
- Q5:From the point P(-2,4), if PQ is drawn perpendicular to the line 7x - 24 y + 10 = 0, find the equation of the line PQ. Also determine the length of PQ. AnswerTo Find : Equation of PQ & Length of PQ Solution :PQ is perpendicular to 7x-24 y+10=0 Slope of given Line = - ( Coefficient of x)/( Coefficient of y) Slope of [7x - 24 y + 10 = 0] = 7/24 Since PQ is perpendicular to 7x - 24 y + 10 = 0 Slope of PQ × Slope of given line = -1 Slope of PQ = -24/7 Also, PQ passes from point P(-2,4) Now , Equation of PQ will be : y -Y1 = m(x-X1) y - 4 = (-24/7)(x-(-2)) y - 4 = (-24/7)(x + 2) 7y - 28 = -24x -48 24x +7y + 20 = 0 distance of a point P(a , b) from a line Ax + By + C = 0 is given by , Distance = |A(a) + B(b) + C|/√(A²+B²) PQ = |7(-2) + (-24)(4) + 10|/√(7²+24²) PQ = | -14 -96 +10 | / √(49+576) PQ = |-100| / √(625) PQ = 100/25 PQ = 4 units Equation of PQ is 24x +7y + 20 = 0 Length of PQ is 4 unit
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