Question

Can you help with Q12?

surface area question

Answer 1

`71.88`

Explanation:

This one's a bit hard to type out.

First, I'll calculate the base. Assuming the bottom isn't included, each surface on the base is calculated with the classic formula, `A=l*w`, and since there are four sides to the base, we'll multiply our answer by `4`.

`SA=4l*w`

We'll plug in the values for length and width, `1.1` and `4.2` respectively.

`SA=4(1.1)(4.2)`

Let's plug this into our handy calculator and get

`SA=18.48`

But that's not all. Next, we calculate the surface area of the top. This is where is gets tricky.

The top is a pyramid, made up of four triangles. Each triangle base is `4.2`.

Now, imagine a dotted line falling from the very tip of the pyramid down to the middle of the base. This dotted line is now the leg of an imaginary right triangle. The length of this dotted line is `6`, which is given to us in the problem. Now, a base is made connecting to this dotted line with a length of `2.1`, half the base length. The hypotenuse of this imaginary triangle is the slope of the pyramid.

Still with me? It's hard to imagine without a picture.

By calculating the hypotenuse of this triangle, we get the slope of the entire pyramid. We'll use the Pythagorean theorem, `a^2 + b^2 = c^2`

`a=6`
`b=2.1`

`(6)^2 + (2.1)^2 = c^2`
`c=sqrt[(6)^2 + (2.1)^2]`

Plug that into your calculator and get

`c=sqrt40.41`

We'll leave it as that for now.

Now we have the slope of the pyramid, which is the height of the pyramid's triangle faces.

The area of a triangle is `a=(b*h)/2`

`h=sqrt40.41`
`b=4.2`

There are four triangles, so we will multiple this by `4` to get surface area.

`SA = 4(b*h)/2 = 2(b*h)`
`SA = 2sqrt40.41*4.2`

Now, let's plug that into our calculator.

Our rounded answer is

`SA = 53.40`

To finish, we add both surface areas we found together.

`SA` Pyramid `= 53.40`
`SA` Base `=18.48`
`SA` Total `= 71.88`