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Evaluate ∫∫∫s z^2 dx dy dz where S is the solid region between the spheres ρ =1and ρ = 2 by using spherical coordinates.
Evaluate D∫∫(x+2y) da, where D is the region bounded by the parabolas y=2x^2 and y= 1+x^2.
Check if the following integrals are independent of path and evaluate those which are independent. i) ∫ (0,0) to (3,4) (6xy-y^3)DX +(3x^2-x^3y)dy. 2) integration (-1,4) to (3,8) (3x^2-2y^2)dx-4xy dy
Find the moment of inertia I2 for the solid above the xy-plane bounded by the paraboloid z=x^2+y^2 and the cylinder x^2+y^2 =9 assuming the mean density to be constant C.
A paint factory produces both interior and exterior house paints for wholesale distribution. Two basic raw materials, A and B, are used to manufacture the paints. The maximum availability of A is 6 tonnes a day and that of B is 8 tonnes a day. The requirements of raw materials per tonne of interior and exterior paints are summarized:
Raw material = A & B
Tonnes of raw material per ton of paint; Exterior, A = 2, B = 1 & Interior, A = 3, B = 3
Maximum availability (Tonnes): A = 4, B = 5
A market survey has established that the daily demand for interior paint cannot exceed that of exterior paint by more than one tonne. The survey also shows that the maximum demand for interior paint is limited to 3 tonnes daily. The whole sale price per tonne is Rs.2000/- for exterior paint and Rs.1000/- for the interior paint. How much of interior and exterior paints should the company produce daily to maximize the gross income? Formulate the problem as an LPP.
Raw material = A & B
Tonnes of raw material per ton of paint; Exterior, A = 2, B = 1 & Interior, A = 3, B = 3
Maximum availability (Tonnes): A = 4, B = 5
A market survey has established that the daily demand for interior paint cannot exceed that of exterior paint by more than one tonne. The survey also shows that the maximum demand for interior paint is limited to 3 tonnes daily. The whole sale price per tonne is Rs.2000/- for exterior paint and Rs.1000/- for the interior paint. How much of interior and exterior paints should the company produce daily to maximize the gross income? Formulate the problem as an LPP.
A torus (a dough nut-shaped object) is formed by revolving the circle
x^2 + y^2 = a^2
about the vertical line
x = b,
where 0 < a < b. Find its volume.
x^2 + y^2 = a^2
about the vertical line
x = b,
where 0 < a < b. Find its volume.
A simple random sample of 36 cans of regular Coke has a mean volume of 12.19 oz. Assume that the standard deviation of all cans of regular Coke is 0.11 oz. Use a 0.01 significance level to test the claim that cans of regular Coke have volumes with a mean of 12 oz, as stated on the label.
a. State the null (H0) and alternative (H1) hypotheses.
a. State the null (H0) and alternative (H1) hypotheses.
High-tex Engineering Company Limited wishes to set flexible budgets for each of its operating departments. A separate maintenance department performs all routine and major repair works on the company’s equipment and facilities. The company has determined that maintenance department performs all routine and major repair works on the company’s equipment and facilities. The company has determined that maintenance cost is primarily a function of machine hours worked in the various production departments.
The maintenance cost incurred and the actual machine hours worked during the months of January, February, March and April 2003 were as follows:
Required:
a) Determine the cost estimation function using:
i High-low method. (2 marks)
ii Regression analysis (2 marks)
b) Using the regression function estimate:
i The maintenance costs that would have been incurred if the machine hours were expected to be 900 in the month of May 2003. (1 mark)
The maintenance cost incurred and the actual machine hours worked during the months of January, February, March and April 2003 were as follows:
Required:
a) Determine the cost estimation function using:
i High-low method. (2 marks)
ii Regression analysis (2 marks)
b) Using the regression function estimate:
i The maintenance costs that would have been incurred if the machine hours were expected to be 900 in the month of May 2003. (1 mark)
Watt Lovell Ltd. (WLL) is trying to decide whether or not to drill for oil on a particular site in North Eastern Kenya. The Chief Engineer has assessed the probabilities that there will be oil as follow, based on past experience.Oil 0.2, No oil 0.8
It is possible for WLL to hire a firm of international consultants to carry out a complete survey of the site. WLL has used the firm many times before and has made the following estimates:
1. If there really is oil, then there is a 95% chance that the report will be favourable.
2. If there is no oil then there is only a 10% chance that the report will indicate that there is oil.
The following additional information is also provided:
· The cost of drilling is Sh.10 million.
· The value of the benefits if oil is found is Sh.70 million
· The cost of obtaining information is Sh.3 million.
a) Advise the company on whether to acquire additional information from the consultants
b) Compute the value of imperfect information. (2 marks)
It is possible for WLL to hire a firm of international consultants to carry out a complete survey of the site. WLL has used the firm many times before and has made the following estimates:
1. If there really is oil, then there is a 95% chance that the report will be favourable.
2. If there is no oil then there is only a 10% chance that the report will indicate that there is oil.
The following additional information is also provided:
· The cost of drilling is Sh.10 million.
· The value of the benefits if oil is found is Sh.70 million
· The cost of obtaining information is Sh.3 million.
a) Advise the company on whether to acquire additional information from the consultants
b) Compute the value of imperfect information. (2 marks)
Find the centre of gravity of a mass in the shape of a semicircular disc of radius 4, if the density at (x,y) is 2y/x^2+y^2.
