Calculate the density of a mixture given the densities and volumes of its components.

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Question 1:
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Question 2: Calculate the density of the mixture.

Calculate the mass of acid.
$\text{Mass of acid} = \text{Density of acid} \times \text{Volume of acid}$ $\text{Mass of acid} = 1.08 \text{ g/cm}^3 \times 200 \text{ cm}^3 = 216 \text{ g}$
Calculate the mass of alcohol.
$\text{Mass of alcohol} = \text{Density of alcohol} \times \text{Volume of alcohol}$ $\text{Mass of alcohol} = 0.8 \text{ g/cm}^3 \times 300 \text{ cm}^3 = 240 \text{ g}$
Calculate the total mass of the mixture.
$\text{Total mass} = \text{Mass of acid} + \text{Mass of alcohol}$ $\text{Total mass} = 216 \text{ g} + 240 \text{ g} = 456 \text{ g}$
Calculate the total volume of the mixture.
$\text{Total volume} = \text{Volume of acid} + \text{Volume of alcohol}$ $\text{Total volume} = 200 \text{ cm}^3 + 300 \text{ cm}^3 = 500 \text{ cm}^3$
Calculate the density of the mixture.
$\text{Density of mixture} = \frac{\text{Total mass}}{\text{Total volume}}$ $\text{Density of mixture} = \frac{456 \text{ g}}{500 \text{ cm}^3} = 0.912 \text{ g/cm}^3$
The density of the mixture is $\boxed{0.912 \text{ g/cm}^3}$ .

Question 3: Find the magnitude of vector OM.
Find the coordinates of point M.
Point M divides line PQ in the ratio 2:1. Given P(5, 1) and Q(11, 4).
Using the section formula $M = \left(\frac{nx_1 + mx_2}{m+n}, \frac{ny_1 + my_2}{m+n}\right)$ , where $(x_1, y_1) = (5, 1)$ , $(x_2, y_2) = (11, 4)$ , $m=2$ , $n=1$ .
$M_x = \frac{1 \times 5 + 2 \times 11}{2+1} = \frac{5 + 22}{3} = \frac{27}{3} = 9$ $M_y = \frac{1 \times 1 + 2 \times 4}{2+1} = \frac{1 + 8}{3} = \frac{9}{3} = 3$
So, the coordinates of M are $(9, 3)$ .
Determine the vector OM.
The vector OM is the position vector of M, which is $\begin{pmatrix} 9 \\ 3 \end{pmatrix}$ .
Calculate the magnitude of vector OM.
$|\vec{OM}| = \sqrt{9^2 + 3^2}$ $|\vec{OM}| = \sqrt{81 + 9}$ $|\vec{OM}| = \sqrt{90}$ $|\vec{OM}| = \sqrt{9 \times 10} = 3\sqrt{10}$
The magnitude of vector OM is $\boxed{3\sqrt{10}}$ .

Question 4:
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Question 5:
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Question 6: A line passes through points (2, 5) and has a gradient of 2.

a) Determine its equation in the form $y=mx+c$ .
Use the given gradient $m=2$ and the point $(x,y) = (2,5)$ in the equation $y=mx+c$ .
$5 = 2(2) + c$ $5 = 4 + c$ $c = 5 - 4$ $c = 1$
Write the equation of the line.
$y = 2x + 1$
The equation of the line is $\boxed{y = 2x + 1}$ .

b) Find the angle it makes with x-axis.
Use the relationship between gradient and angle.
The gradient $m = \tan \theta$ , where $\theta$ is the angle the line makes with the positive x-axis.
$\tan \theta = 2$ $\theta = \arctan(2)$ $\theta \approx 63.43^\circ$
The angle the line makes with the x-axis is $\boxed{63.43^\circ}$ (to two decimal places).

Question 7: Find the value of P when Q=5.
Write the general equation for P.
P is partly constant and partly varies as the cube of Q.
$P = k + cQ^3$
where $k$ and $c$ are constants.
Use the given conditions to form a system of equations.
When $Q=1, P=23$ :
$23 = k + c(1)^3 \implies 23 = k + c \quad \text{(Equation 1)}$
When $Q=2, P=44$ :
$44 = k + c(2)^3 \implies 44 = k + 8c \quad \text{(Equation 2)}$
Solve the system of equations for $k$ and $c$ .
Subtract Equation 1 from Equation 2:
$(44 - 23) = (k + 8c) - (k + c)$ $21 = 7c$ $c = \frac{21}{7} = 3$
Substitute $c=3$ into Equation 1:
$23 = k + 3$ $k = 23 - 3 = 20$
Write the specific equation for P.
$P = 20 + 3Q^3$
Find P when Q=5.
$P = 20 + 3(5)^3$ $P = 20 + 3(125)$ $P = 20 + 375$ $P = 395$
The value of P when Q=5 is $\boxed{395}$ .

Question 8: Determine the translation vector T and find the coordinates of A' and B'.
Given vertices A(1, 2), B(3, 5), C(4, 1). Image of C is C'(6, -2).

a) Determine the translation vector T.
Use the formula Image = Original + Translation Vector.
$C' = C + T$ $\begin{pmatrix} 6 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + T$ $T = \begin{pmatrix} 6 \\ -2 \end{pmatrix} - \begin{pmatrix} 4 \\ 1 \end{pmatrix}$ $T = \begin{pmatrix} 6-4 \\ -2-1 \end{pmatrix} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$
The translation vector is $\boxed{T = \begin{pmatrix} 2 \\ -3 \end{pmatrix}}$ .

b) Find the co-ordinates of A' and B' under the translation vector T.
Find A'.
$A' = A + T$ $A' = \begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 1+2 \\ 2-3 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$
So, A' is $(3, -1)$ .
Find B'.
$B' = B + T$ $B' = \begin{pmatrix} 3 \\ 5 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 3+2 \\ 5-3 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$
So, B' is $(5, 2)$ .
The coordinates are $\boxed{A'(3, -1)}$

Calculate the density of a mixture given the densities and volumes of its components.

Step-by-step solution

Calculate the mass of acid.

Calculate the mass of alcohol.

Calculate the total mass of the mixture.

Calculate the total volume of the mixture.

Calculate the density of the mixture.

Find the coordinates of point M.

Determine the vector OM.

Calculate the magnitude of vector OM.

Use the given gradient $m=2$ and the point $(x,y) = (2,5)$ in the equation $y=mx+c$ .

Write the equation of the line.

Use the relationship between gradient and angle.

Write the general equation for P.

Use the given conditions to form a system of equations.

Solve the system of equations for $k$ and $c$ .

Write the specific equation for P.

Find P when Q=5.

Use the formula Image = Original + Translation Vector.

Find A'.

Find B'.

Still stuck on this one?

More Chemistry Questions

Calculate the density of a mixture given the densities and volumes of its components.

Step-by-step solution

Calculate the mass of acid.

Calculate the mass of alcohol.

Calculate the total mass of the mixture.

Calculate the total volume of the mixture.

Calculate the density of the mixture.

Find the coordinates of point M.

Determine the vector OM.

Calculate the magnitude of vector OM.

Use the given gradient m=2m=2m=2 and the point (x,y)=(2,5)(x,y) = (2,5)(x,y)=(2,5) in the equation y=mx+cy=mx+cy=mx+c.

Write the equation of the line.

Use the relationship between gradient and angle.

Write the general equation for P.

Use the given conditions to form a system of equations.

Solve the system of equations for kkk and ccc.

Write the specific equation for P.

Find P when Q=5.

Use the formula Image = Original + Translation Vector.

Find A'.

Find B'.

Still stuck on this one?

More Chemistry Questions

Use the given gradient $m=2$ and the point $(x,y) = (2,5)$ in the equation $y=mx+c$ .

Solve the system of equations for $k$ and $c$ .