Determine the equation of the tangent to the circle with equation (x−3)2 + (y−5)2 = 5 at the point (6, 7).

The equation of the circle is $(x-3)^2 + (y-5)^2 = 5$.
The center of the circle is $(h, k) = (3, 5)$.

The point of tangency is $(x_1, y_1) = (6, 7)$.
The center is $(x_c, y_c) = (3, 5)$.
The gradient of the radius $m_r$ is:

The tangent line is perpendicular to the radius at the point of tangency. If $m_t$ is the gradient of the tangent, then $m_t \times m_r = -1$.

Using the point $(6, 7)$ and the gradient $m_t = -\frac{3}{2}$:

Multiply both sides by 2 to eliminate the fraction:

Rearrange the equation into the form $Ax + By + C = 0$:

The equation of the tangent is: $\boxed{3x + 2y - 32 = 0}$

11. The volume of a cylinder is given by $V = \pi r^2 h$. Find the percentage change in $V$ if $r$ increases by 8% and $h$ decreases by 12%.

Let the original radius be $r_0$ and the original height be $h_0$.
Original volume $V_0 = \pi r_0^2 h_0$.

Radius increases by 8%:
New radius $r_1 = r_0 + 0.08 r_0 = 1.08 r_0$.

Height decreases by 12%:
New height $h_1 = h_0 - 0.12 h_0 = 0.88 h_0$.

Substitute the expressions for $r_1$ and $h_1$:

Since $V_0 = \pi r_0^2 h_0$, we have:

Percentage change $= \frac{V_1 - V_0}{V_0} \times 100\%$

The percentage change in volume is: $\boxed{2.64\% \text{ increase}}$ (rounded to two decimal places).

12. A trader deals in two types of rice; rice A and rice B. Type A rice costs Kshs. 400 per bag and type B costs Kshs. 350 per bag. The trader mixes 30 bags of type A with 50 bags of type B. If he sells the mixture at a profit of 20%, calculate the selling price of one bag of the mixture.

Cost of 30 bags of type A = $30 \text{ bags} \times \text{Kshs. } 400/\text{bag} = \text{Kshs. } 12000$.

Cost of 50 bags of type B = $50 \text{ bags} \times \text{Kshs. } 350/\text{bag} = \text{Kshs. } 17500$.

Total cost of mixture = Cost of type A + Cost of type B
Total cost of mixture = $\text{Kshs. } 12000 + \text{Kshs. } 17500 = \text{Kshs. } 29500$.

Total bags = 30 bags (Type A) + 50 bags (Type B) = 80 bags.

Cost price per bag = $\frac{\text{Total cost of mixture}}{\text{Total bags}} = \frac{\text{Kshs. } 29500}{80} = \text{Kshs. } 368.75$.

Selling price per bag = Cost price per bag $\times (1 + \text{Profit percentage})$
Selling price per bag = $\text{Kshs. } 368.75 \times (1 + 0.20)$
Selling price per bag = $\text{Kshs. } 368.75 \times 1.20$
Selling price per bag = $\text{Kshs. } 442.50$.
The selling price of one bag of the mixture is: $\boxed{\text{Kshs. } 442.50}$