A manufacturing company produces electronic components for various devices.
They are analyzing the production data for the past 3 months, which includes quantities of components produced and the corresponding costs. The data is presented in the table below: Express the total quantity produced from May to July in standard form correct to 3 significant figures. Calculate the average cost per unit for these 3 months. Express your answer in cents. Answer ... cents Given that the percentage increase in the quantity produced from July to August is 11.8%, calculate the quantity produced in August. Leave your answer in standard form correct to 3 significant figures. Answer ... units The company wants to build a prototype of a particular electronic component they are manufacturing. The radius of the actual electronic component is $3.8\times10^{-5}$ m. In a scale drawing, the radius of the prototype of the electronic component is 1.9 cm. Find the scale used for the drawing. Give your answer in the form n : 1. Answer ... : 1 Given that the prototype has a total surface area of $1.81\times10^{-3}$ m $^2$ , find, in cm $^2$ , the actual total surface area of the electronic component. Give your answer in standard form. Answer ... cm $^2$

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Worked solution

2(a)(i)

1. Add the quantities produced in May, June and July.

$5.8\times10^5+4.3\times10^6+7.6\times10^5=580000+4300000+760000=5640000$

2. Express the total in standard form correct to 3 significant figures.

$5640000=5.64\times10^6$
This is already correct to 3 significant figures.

Answer

$5.64\times10^{6}$ units

2(a)(ii)

1. Calculate the total production cost for the 3 months.

\begin{aligned} \text{Total cost} &= (5.8\times10^5)(0.0211) +(4.3\times10^6)(0.0183) +(7.6\times10^5)(0.0203)\\ &= 12238+78690+15428\\ &= 106356 \end{aligned}

2. Divide by the total quantity and convert the average cost from dollars to cents.

Using the total quantity from part (i),
$\text{Total quantity}=5.64\times10^6=5640000.$

\begin{aligned} \text{Average cost per unit} &= \frac{106356}{5640000}\\ &=0.0188574\ldots \text{ dollars}\\ &=0.0188574\ldots \times 100\\ &=1.88574\ldots \text{ cents}\\ &\approx 1.89\text{ cents} \end{aligned}

Answer

$1.89$ cents

2(a)(iii)

1. Apply the percentage increase to July's quantity.

July quantity $=7.6\times10^{5}$ units.

An increase of $11.8\%$ means multiply by
$(1+\frac{11.8}{100})=1.118.$

August quantity
$=(7.6\times10^{5})(1.118)=849680.$

2. Express the August quantity in standard form correct to 3 significant figures.

$849680=8.4968\times10^{5}$

Correct to 3 significant figures,
$8.4968\times10^{5}=8.50\times10^{5}.$

Answer

$8.50\times10^{5}$ units

2(b)(i)

1. Convert the actual radius from metres to centimetres.

Actual radius
$=3.8\times10^{-5}\text{ m}$

Since $1\text{ m}=100\text{ cm}$ ,
$3.8\times10^{-5}\times100=3.8\times10^{-3}\text{ cm}$

2. Find the scale in the form drawing radius : actual radius.

Scale
$=1.9:3.8\times10^{-3}$

Divide both sides by $3.8\times10^{-3}$ :
$\frac{1.9}{3.8\times10^{-3}}:1=500:1$

Answer

$500 : 1$

2(b)(ii)

1. Convert the prototype surface area from square metres to square centimetres.

Since $1\text{ m}^2=10\,000\text{ cm}^2$ ,
$1.81\times10^{-3}\text{ m}^2=1.81\times10^{-3}\times10^4=18.1\text{ cm}^2.$

2. Use the area scale factor to find the actual surface area.

From part (b)(i), the linear scale is $500:1$ .

Hence the area scale is
$500^2:1^2=250\,000:1.$

So the actual total surface area is
$\frac{18.1}{250\,000}=0.0000724=7.24\times10^{-5}\text{ cm}^2.$

Answer

$7.24\times10^{-4}\text{ cm}^2$

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Worked solution

2(a)(i)

2(a)(ii)

2(a)(iii)

2(b)(i)

2(b)(ii)

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