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Substitute values into formulae for the area or volume of various geometric objects.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Answer the following questions by substituting the correct values into the given equations.

When inserting numbers into your calculator, make sure that you place brackets correctly.

\n

#### a)

\n

We can see from the diagram that the radius of the frisbee is $\\var{mccall[2]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the area of a circle with $\\var{mccall[2]}$ gives,

\n

\\begin{align}
\\mathrm{Area} &= \\pi r^2 \\\\
&= \\pi\\times(\\var{mccall[2]})^2 \\\\
&= \\var{dpformat((mccall[2])^2, 2)}\\pi\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(pi *(mccall[2])^2, 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}

\n

#### b)

\n

We can see from the diagram that the triangle has two sides with lengths $\\var{length_cdp2}$ $\\mathrm{cm}$, $\\var{length_bdp2}$ $\\mathrm{cm}$ and an angle $\\var{c_thetadp2}\\mathrm{°}$ .
Replacing the letters $a$, $b$ and $C$ in the formula for the area of a triangle with $\\var{length_cdp2}$, $\\var{length_bdp2}$ and $\\var{c_thetadp2}$ respectively gives,

\n

\n

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2}ab\\sin{C} \\\\
&= \\frac{1}{2} \\times \\var{length_cdp2} \\times \\var{length_bdp2} \\times \\sin(\\var{c_thetadp2}) \\\\
&= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2* pi/180), 5)}\\, \\mathrm{cm}^2  \\\\
\\end{align}

\n

\n

#### c)

\n

We can see from the diagram that the radius of the cone is $\\var{r}$ $\\mathrm{cm}$ and the height is $\\var{h}$ $\\mathrm{cm}$.
Replacing the letters $r$ and $h$ in the formula for the volume of a cone with $\\var{r}$ $\\mathrm{cm}$ and $\\var{h}$ $\\mathrm{cm}$ respectively gives,

\n

\\begin{align}
\\mathrm{Volume} &= \\frac{h}{3} \\pi r^2 \\\\
&= \\frac{(\\var{h})}{3} \\times \\pi \\times (\\var{r})^2 \\\\
&= \\var{dpformat((pi)*(h/3)*(r)^2 , 5)}\\, \\mathrm{cm}^3 \\\\
&=\\var{dpformat(h/3 * pi * (r)^2, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\n

#### d)

\n

We can see from the diagram that the radius of the tennis ball is $\\var{mccall[1]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the volume of a sphere with $\\var{mccall[1]}$ gives,

\n

\\begin{align}
\\mathrm{Volume} &= \\frac{4}{3} \\pi r^3 \\\\
&= \\frac{(4)}{(3)} \\times \\pi \\times (\\var{mccall[1]})^3 \\\\
&= \\var{dpformat((4/3)*pi*mccall[1]^3, 5)}\\,  \\mathrm{cm}^3 \\\\
&= \\var{precround(((4/3)* pi) *(mccall[1])^3, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\n

#### e)

\n

We can see from the diagram that the trapezium has two parallel sides with length $\\var{trap_length_a}$ $\\mathrm{cm}$, $\\var{trap_length_b}$ $\\mathrm{cm}$ and height $\\var{trap_h}$ $\\mathrm{cm}$.
Replacing the letters $a$, $b$ and $h$ in the formula for the area of a trapezium with $\\var{trap_length_a}$, $\\var{trap_length_b}$ and $\\var{trap_h}$ respectively gives,

\n

\n

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5)(trap_length_a +trap_length_b) trap_h, 2)}\\, \\mathrm{cm}^2 \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

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Rounded value for the length of c.

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Defines the point for the height of the trapezium.

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A random variable which will be inputted by the student.

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The constant coefficient

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List of names to randomise. Can change to any name inserted

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Rounded value for the length of b.

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For triangle - The length of the vector BC

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Rounded theta value.

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For triangle - The length of the vector AC

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Position of point B in Geogebra. This point is randomised to make the triangles different.

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Defines the pronoun in the question.

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Definition of the points to put into Geogebra

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The height for volume of a cone.

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Height of the trapezium

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This calculates the area of the triangle for part b)

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Position of the point A in Geogebra. This point is fixed so the triangle doesn't hang in one corner or the whole page.

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n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.

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List of names to randomise. Can change to any name inserted

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A random number to define the height of the trapezium.

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Theta is randomised by the lengths

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Calculates the area of the trapezium

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Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.

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For triangle - The length of the vector AB

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Creates the point D on the trapezium

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Triangle - A variable point which ultimately decides how the triangle looks.

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Creates the point A on the trapezium

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Creates the point C on the trapezium

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Matrix of random variables used to create length in the questions.

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The x^2 coefficient

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Creates the point B on the trapezium

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The x coefficient

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Calculate the area of a frisbee, assuming that the frisbee can be modelled as a circle, given the formula for the area of a circle is

\n

\$\\mathrm{Area} = \\pi r^2.\$

\n

\n

$\\mathrm{Area}$ = [[0]] $\\mathrm{cm}^2$    Round your answer to 2 decimal places.

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Calculate the area of the triangle given that the area of any triangle can be calculated using the formula

\n

\$\\mathrm{Area} = \\frac{1}{2}ab\\sin{C}.\$

\n

{geogebra_applet('https://www.geogebra.org/m/jcUJu6F4',defs)}

\n

All lengths are in centimetres.

\n

$\\mathrm{Area} =$ [[0]] $\\mathrm{cm}^2$   Round your answer to 2 decimal places.

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Calculate the volume of a cone given the formula for the volume of a cone is

\n

\$\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\$

\n

\n

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$   Round your answer to 1 decimal place.

\n

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{name[n]} has a tennis ball and {pronoun} wants to find the volume of the ball. Using the diagram and the formula for the volume of a sphere, calculate the volume of the ball.

\n

\$\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\$

\n

\n

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$   Round your answer to 1 decimal place.

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Find the area of the trapezium given the formula for the area of a trapezium is

\n

\$\\mathrm{Area} = \\frac{1}{2}(a+b) h .\$

\n

{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}

\n

\n

All lengths are given in metres.

\n

$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$   Round your answer to 1 decimal place.

\n

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