1:2:1 Maths Tuition

Watch The Bubble Grow.....

Query:

The diameter of a sphere is increased by 5%. Find the percentage increase in the surface area of the sphere.

Response:

 

An Annulus Problem....

Query: Two concentric circles have radii of r cm and 6 cm (where r>6). If the area between the circumferences is 64π, find the size of the radius r.

Response:

Estmating Height...

Query: How do I estimate the height of this mast (see diagram)?

Response: The man's height has to be estimated first. Next, you calculate how many men of the same height (i.e. standing on each other's heads) it would take to reach the top of the mast.

Click on the graphic below to see how you would show this.

Rotational Symmetry...

Query: What is the order of rotational symmetry of this shape?

Response: If you rotate the shape you can see that it will fit its own outline four times in each complete revolution.

Hence the order of rotational symmetry is 4.

Tricky Area Question...

Query: How do I find the shaded area, given that the circular shapes are quarters (i.e. quadrants) of circles whose radii of 12 cm. are equal to the side length of the square ABCD, with centres A and C.

Response: Please click on the graphic below to view the solution.

Rotational Symmetry...

Query: What is the order of rotational symmetry of this shape?

Response: If you rotate the shape you can see that it will fit its own outline twice in each complete revolution.

Hence the order of rotational symmetry is 2.

How Many Cuppas?

Query: A tea urn is shaped like a cylinder with a circular cross-section of diameter 35cm and height 60cm. It is filled with tea. How many cups of capacity 150 cubic cms. can be served from the urn?

Response: The answer is 384 cups. Please click on the graphic below to view the solution.

 

Tetrahedrons...

Query [from Loopy]: I get stuck on questions like this on similar shapes. How do I do it?

   

Response: Take a look at the solution provided (opens in new window):

Heavy Metal...

Query [From LAL]: A 20cm square sheet of metal has equal quadrants cut from the corners, as shown in the diagram. The area left is 246 sq.cm and we need to find the radius of the circle from which the quadrants are taken. [use pi as 22/7]

Response:

• Area of the square is 20x20=400 sq cm
• Take away area that remains
• Quadrants add up to 154 sq cm.
• Four quadrants equal 1 circle
• Calculate the radius of this circle.

The answer is 7 cm.

Why Is This Correct?

Query [from Aristotle]: There is a missing shape from this series...

 

The correct answer is below - can you explain why this is?

Response: We sense we're being tested here -  so here goes....!

Take a look at the rows and columns of the original three by three shapes (minus the missing one, that is) and try to spot the following clues...

• In each row there is one occurrence of the outside shape
• In each column there is one occurrence of the outside shape
• Each outline shape contains the same 3 mini-shapes (star, circle & triangle)
• Within each outline shape there are 3 mini-shapes (1 black and 2 clear)
• Each column has the same mini-shape coloured in black
• Each row has a different mini-shape coloured in black

On The Hoof...

Query [from Ron Gale]: I run a ranch in Canada, where we train horses. I would like to know how many cubic yards of sand I will need for my arena measuring 100yds. X 300yds. (with rounded ends). Also, how do I calculate the cubic yards per inch of depth?

Response: Initially, you should disregard the effect of the rounded corners, as it will probably only actually represent a very small percentage of the top surface area.

A depth of 1 yd. of sand will require a volume of 100x300x1 cu.yds.

So, a 1" depth requires 1/36^th of this, i.e. (100x300x1) ÷ 36 = 833.3 cu.yds.

If you wanted an 8" depth, for example, then simply multiply this figure by x8 (to get an approximate total of 6666 cu.yds., to the nearest cu.yd.).

At this stage, you should make a decision about the likely effects of the rounded corners. So, if the radius of these corners is less than 1 yd., you could just ignore it.

However, if the corners are very rounded then apply a 2.5% or a 5% (or even higher) reduction to the overall volume you have calculated, i.e. multiply your total by x0.975 or x0.95.

COMMENT [from DH]: You wouldn't believe how nice it is to work in imperial units once more!

1:2:1 Challenge (Solution 07/03)...

The answer is 9 cm. These are the steps you need to take....

1. Work out how much water, V cm³ is contained in the hemisphere
2. Treat the cylinder as a prism and work out area of circular cross-section
3. Apply the formula: 'Volume' = 'Cross-Sectional Area' x h
4. Equate the Volumes calculated in steps 1. and 3. (since no water was lost).
5. Solve for h.

Download Solution [pdf]

1:2:1 Challenge (Problem 07/03)...

A hemispherical bowl of internal diameter 12 cm. is full of water and all the water is emptied into a cylindrical can of base diameter 8 cm.

How deep will the water level in the cylinder become?

Please e-mail us your solution - you can attach a scan of your handwritten version, if it's more convenient. The solution will be posted at 12 noon [GMT] on Friday, 14/09/07.

Again, no prizes on offer - just the pleasure of being first to solve it!  But can you solve it?

They Look Similar...

Query [from KP]: I've been given two triangles - one with sides of 6, 8 and 12 cm and the other with sides 9, 10 and 15 cm. I'm meant to prove they are not similar?

Response: Although the triangles in the diagram certainly look similar, they are not drawn to scale. You will need to compare corresponding sides to test for similarity, by seeing if one triangle is a true enlargement of the other.

• Smallest (6 cm : 9 cm)
• Middle (8 cm : 10 cm)
• Longest (12 cm : 15 cm).

In their lowest terms: Smallest (2:3), Middle (4:5) & Longest (4:5).

Since these ratios are not identical, it follows that the triangles are not similar.

Absurd Angles...

Query [from CarolK]: I've been given a triangle with angles of 30, 60 and 90 degrees. The hypotenuse length is 3 point 5 and I'm meant to find the length opposite the 60 degree angle in SURD form? I could do it using a calculator easily enough!

Response: All 30-60-90 triangles are enlargements of the one shown in the diagram [this should really be memorised].

In this case, comparing the lengths of both hypotenuses gives an enlargement ratio of 2:3.5, i.e. a scale factor of 1.75.

This means that the side opposite the 60° becomes 1.75 x √3.

You should now completely rationalise this surd term by removing the decimals.

View the answer [pdf]

Similar Polygons...

Most students find the whole topic of similarity very challenging.

At its most simple, two figures (2 or 3 dimensions) are said to be similar if one is an enlargement of the other. This means that the angles associated with both shapes are identical and the lengths are in a fixed ratio, sometimes called the scale factor.

Query [from TerryL]: I've been given a diagram with two regular pentagons and told that the area of the inner pentagon is 5cm². How do I find the shaded area?

Response: As both pentagons are regular we know immediately that they must be similar to each other.

The length scale factor of enlargement is x 2, worked out by comparing the lengths of y and 2y (since one is 2 'times' the other).

This means that the area scale factor becomes x 4, which was arrived at by taking the square of the length scale factor.

In this case, therefore the outer area must be 20cm² , which leaves the shaded area as 15cm².

Prism...

A prism is defined as a three-dimensional object which has an uniform cross-sectional area running along one axis. So, if the prism was cut perpendicular to this axis the new faces revealed would be exactly the same size and shape as the ends of the original object. Confusing, isn't it?

Query: You're given a pentagonal prism with a length of 2.5 cm. and a volume of 1.875 cubic cm. and you're meant to find the area of cross-section?

Response: The shaded area here refers to the uniform cross-section, which all prisms must have.

The Volume of any prism = 'Cross-Sectional Area (shaded)' x 'Length'

So the Cross-Sectional Area = 1.875 ÷ 2.5 = 0.75 cm².

Half A Circle?

This is a type of problem that we've had to explain many times in our teaching careers!

Query [from AL]: I was asked to find the perimeter of a semi-circular shape of diameter 20 cm. I worked out the circumference of the circle and then divided it by two. I got this question wrong, why?

Response: It is true that the circular part of the perimeter of the semicircle is exactly half the circumference of the whole circle.

However, a journey around any perimeter involves following the outline of the shape until you end up back where you started!

You forgot to add on the diameter, which joins the opposite ends of the semicircle, when calculating the whole perimeter.

Perimeter = ½ x (2 x π x 10) + 20 ≈ 51.4 cm.

Scatter Graph...

Drawing a line of best fit for data usually causes real problems for students.

At GCSE (11-16) level, however, you just have to draw your best 'guesstimate', as there is no elaborate theory of 'linear regression' to apply - this is only studied for Advanced, Higher or University courses!

Query [from GOG]: In this scatter graph, I'm given the mean point and asked to draw a line of best fit?

Response: A line of best fit should pass through the coordinates of the mean points of the two 'graphed' variables and it is then drawn 'by eye' (that is, trying to get about as many points below as above!)

Here, we suggest you plot the given mean point and then using your ruler sideways (thin side down) you

• Make sure the ruler goes through the mean point
• Keep adjusting the direction until it looks about right.
• Mark the best position & Draw the line.

NOTE: If the mean isn't given and the question is of low value (only worth 1 or 2 marks), then just draw it by 'eye', without calculating the means.

Download Solution [pdf]

It's Pythagoras, Really...

Query [from PaulG]: I've been asked to find the distance from the point A(2,3) to another point B(5,-4) and I get the answer to be the square root of 10 when I apply the distance formula? What have I done wrong?

Response: The distance formula is actually an application of the Theorem of Pythagoras. If you draw a quick sketch of these points onto an X-Y coordinate grid, you'll get the picture that we've created here...

The X_AB shift is +3 units and the Y_AB shift is -7 units (not -1)

This means that AB² = 3² + (-7)² = 58, giving AB = √58.

Annulus Horrible?

Query [from DD007]: What's an annulus and how do you calculate the area?

Response: This diagram shows a shaded-in picture of an annulus, marked with an internal radius of 3 and an external radius 5.

NOTE: The two circles which make up the annulus are concentric, which means that they share the same centre.

Download Worked Example [pdf]

Angles In Polygons...

Query [from KL007]: I've been told that if you add up all the exterior angles of a regular polygon they add up to 360 degrees. How does this help you find the interior angles of a decagon?

Response: Calculating the exterior angles first is a very useful technique to quickly calculate the interior angles of any regular polygon.

Download Theory & Solution [pdf]

Spot The Hypotenuse...

Query [from PL]: I've been given the lengths of the sides of a right-angled triangle as three algebra terms, 3x, x+1 & 2x+5 (longest).  What do I do next?

Response: This is a disguised question which requires the application of the Theorem of Pythagoras - the clue is in the fact that it is a right-angled triangle and only side lengths are mentioned.

   Download Solution [pdf]

My Hero!

Query [from SCJ]: I know the lengths of the sides of an isosceles triangle and my teacher has shown me how to work out the area of the triangle by splitting it in two and using trigonometry. I wondered how to work out the area for a triangle that isn’t isosceles?

Response: You have to use a formula named after the Greek philosopher Heron (usually referred to as Hero's Formula, in Maths text books - some texts even attribute the formula to Archimedes - so who knows?)

This formula is also very useful for modelling calculations involving maximum area for a fixed perimeter of a triangle whilst using a spreadsheet.

Download Fuller Explanation [pdf]

How to find the angles?

Query [from PLJ]: I've been given a quadrilateral with angles marked as y, 2y, 3y & 4y and I'm meant to find what the value of y is?

Response: The key point here is that all the four angles of any quadrilateral add up to 360º and in this case also add up to 10yº.

Hence, y = 36º is the answer.

Download Solution [pdf]