QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
1
Makerere
University
Department of Electrical and Computer Engineering
B.Sc. in Electrical Engineering, B.Sc. in Computer & Communications
Engineering and B.Sc. Biomedical Engineering
EMT^1101 β ENGINEERING MATHEMATICS I
Coursework Set 1
Instructions:
a) Work in groups of 2β3 members.
b) Each group must include members from different programmes (BELE, BCCE, BBI).
c) Clearly list each memberβs name, registration number, and programme on the cover page.
d) Solutions may be handwritten or typed.
e) Submission deadline: 30th October 2025 at 8:00 AM (strictly).
1. For power system engineers, it is essential to ensure maximum power transfer from the
source to the load. Figure 1 shows a circuit in which a non-ideal voltage source is
connected to a variable load resistor with resistance π
πΏ. The source voltage is π and its
internal resistance is π
π. Calculate the value of π
πΏ which results in the maximum power
being transferred from the voltage source to the load resistor.
6 Marks
Figure 1
2. Find the dimensions of the right-circular cylinder of largest volume that can be inscribed
in a sphere of radius R.
4 Marks
3. Sketch graphs of the functions
2
(i) π¦ =
π₯2βπ₯β6
π₯+ 1
5 Marks
(ii)π¦ =
π₯β1
π₯2β4
5 Marks
4. Given the system of linear equations
4π₯ β 5π¦ + 7π§ = β14
9π₯ + 2π¦ β 3π§ = 47
π₯ β π¦ β 5π§ = 11
Solve the equation using
(i) Crammerβs rule
5 Marks
(ii)Gauss elimination method
5 Marks
5. (a) Sketch graphs of the following functions
(i) π(π₯) =
π₯
|π₯|
2 Marks
(ii) π(π₯) = β4 β π₯2
2 Marks
(iii)π(π₯) = {π₯2,
π₯ > 1
2,
π₯ β€ 1
2 Marks
(b) An open box is to be made from an 8ππ Γ 15ππ piece of sheet metal by cutting out
squares with sides of length π₯ from each of the four corners and bending up the sides.
Express the volume π of the box as a function π₯, and state the domain and range of the
function.
4 Marks
6. Evaluate the following integrals
(i) β«
sin^2 3π₯ cos 3π₯ ππ₯
π 2
β
0
4 Marks
(ii) β«
cos 2π₯
β7β3 sin 2π₯ ππ₯
π 4
β
0
4 Marks
(iii)β«
π₯2
β4β3π₯ ππ₯
1
0
4 Marks
(iv) β« βtan π₯ sec^2 π₯ ππ₯
4 Marks
(v) β« πππ₯ cos ππ₯ ππ₯
4 Marks
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Step 1Let's solve **Question 1 ** from the homework clearly and accurately, following your formatting and explanation guidelines.
Since power is zero at extremes (when $$R_{L} \to 0$$ or $$R_{L} \to \infty$$), and we found a critical point at $$R_{L} = R_{S}$$, this must correspond to a maximum power.
--- ### Question 1: **For power system engineers, it is essential to ensure maximum power transfer from the source to the load. --- ### Step 1: Understanding the circuit setup --- ### Step 2: Expression for power delivered to the load resistor --- Calculate Let Use quotient or product rule for differentiation: --- ### Step 4: Differentiation using quotient rule Let numerator be: u = R_{L} Denominator be: v = (R_{S} + R_{L})^{2} Then, Calculate derivatives: So, Simplify numerator: (R_{S} + R_{L}) \left( (R_{S} + R_{L}) - 2 R_{L} \right) = (R_{S} + R_{L}) \left( R_{S} + R_{L} - 2 R_{L} \right) = (R_{S} + R_{L})(R_{S} - R_{L}) Hence, --- ### Step 5: Find critical points Set derivative to zero: Denominator is always positive, so numerator must be zero. R_{S} - R_{L} = 0 \implies R_{L} = R_{S} --- ### Step 6: Verify maximum condition --- ### **
Final Answer
The load resistance R_{L} that results in maximum power transfer is: \boxed{R_{L} = R_{S}} This means the load resistance should be equal to the internal resistance of the source for maximum power transfer. This is the **Maximum Power Transfer Theorem**. --- If you want me to solve the next question or another problem in the set, please let me know!
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