Define the following terms: (i) measurement (ii) quantity (iii) unit (iv) accuracy (v) precision

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GSE 212: BASIC GENERAL MATHEMATICS III (1C)

Question 1:

a) Define the following terms:

(i) Measurement: The process of assigning a numerical value to a physical quantity using a standard unit.
(ii) Quantity: A property of a phenomenon, body, or substance that can be quantified by measurement.
(iii) Unit: A standard amount of a physical quantity, such as a meter for length or a kilogram for mass, used for comparison.
(iv) Accuracy: How close a measured value is to the true or actual value of the quantity being measured.
(v) Precision: The degree to which repeated measurements under unchanged conditions show the same results; it indicates the reproducibility of a measurement.

b) Given $3x + 4y = 26$ , express $x$ in terms of $y$ and find $x$ when $y = 5$ and $y = 10$ .

Express $x$ in terms of $y$ .
Subtract $4y$ from both sides, then divide by 3.
$3x + 4y = 26$ $3x = 26 - 4y$ $x = \frac{26 - 4y}{3}$
The expression for $x$ in terms of $y$ is $\boxed{x = \frac{26 - 4y}{3}}$ .
Find $x$ when $y = 5$ .
Substitute $y = 5$ into the expression for $x$ .
$x = \frac{26 - 4(5)}{3}$ $x = \frac{26 - 20}{3}$ $x = \frac{6}{3}$ $x = 2$
When $y = 5$ , $x = \boxed{2}$ .
Find $x$ when $y = 10$ .
Substitute $y = 10$ into the expression for $x$ .
$x = \frac{26 - 4(10)}{3}$ $x = \frac{26 - 40}{3}$ $x = \frac{-14}{3}$
When $y = 10$ , $x = \boxed{-\frac{14}{3}}$ .

c) In the diagram, PQST is a parallelogram, PR is a straight line. $|TS| = 8 \text{cm}$ , $|SM| = 6 \text{cm}$ and area of triangle PSR $= 36 \text{cm}^2$ . Find the value of $|QR|$ .
Find the length of the base PR using the area of triangle PSR.
The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$ . Here, the base is PR and the height is SM.
$\text{Area of } \triangle PSR = \frac{1}{2} \times |PR| \times |SM|$ $36 \text{ cm}^2 = \frac{1}{2} \times |PR| \times 6 \text{ cm}$ $36 = 3 \times |PR|$ $|PR| = \frac{36}{3}$ $|PR| = 12 \text{ cm}$
Determine the length of PQ.
Since PQST is a parallelogram, opposite sides are equal in length. Therefore, $|PQ| = |TS|$ .
$|PQ| = 8 \text{ cm}$
Calculate the value of $|QR|$.
From the diagram, PR is a straight line, so $|PR| = |PQ| + |QR|$ .
$12 \text{ cm} = 8 \text{ cm} + |QR|$ $|QR| = 12 - 8$ $|QR| = 4 \text{ cm}$
The value of $|QR|$ is $\boxed{4 \text{ cm}}$ .

Question 2:

a) The formula $T = \frac{4b^2}{21}(d - \frac{3}{5}b)$ gives the approximate mass, $T$ tonnes, of a ship $d$ metres long and $b$ metres wide.

i) Make $d$ the subject of the formula.
Multiply both sides by 21.
$21T = 4b^2 \left(d - \frac{3}{5}b\right)$
Divide both sides by $4b^2$ .
$\frac{21T}{4b^2} = d - \frac{3}{5}b$
Add $\frac{3}{5}b$ to both sides.
$d = \frac{21T}{4b^2} + \frac{3}{5}b$
The formula with $d$ as the subject is $\boxed{d = \frac{21T}{4b^2} + \frac{3}{5}b}$ .

ii) Calculate the length of a 4500 tonnes ship if it is 20m wide, giving your answer to the nearest 5m.
Substitute the given values into the formula for $d$ .
Given $T = 4500$ tonnes and $b = 20$ m.
$d = \frac{21(4500)}{4(20)^2} + \frac{3}{5}(20)$ $d = \frac{94500}{4(400)} + \frac{60}{5}$ $d = \frac{94500}{1600} + 12$
Perform the calculations.
$d = 59.0625 + 12$ $d = 71.0625 \text{ m}$
Round the answer to the nearest 5m.
71.0625 m is closer to 70 m than 75 m.
The length of the ship is approximately $\boxed{70 \text{ m}}$ .

b) A wooden structure in form of a cone mounted on a hemispherical base. The vertical height of the cone is 24cm and the base radius 7cm. Calculate, correct to 3 Significant figures, the surface area of the structure. ( $\pi = \frac{22}{7}$ ).
Calculate the slant height ( $l$ ) of the cone.
The slant height can be found using the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$ .
$l = \sqrt{7^2 + 24^2}$ $l = \sqrt{49 + 576}$ $l = \sqrt{625}$ $l = 25 \text{ cm}$
Calculate the curved surface area of the cone ( $CSA_{cone}$ ).
The formula for the curved surface area of a cone is $\pi r l$ .
$CSA_{cone} = \frac{22}{7} \times 7 \text{ cm} \times 25 \text{ cm}$ $CSA_{cone} = 22 \times 25 \text{ cm}^2$ $CSA_{cone} = 550 \text{ cm}^2$
Calculate the curved surface area of the hemisphere ( $CSA_{hemi}$ ).
The formula for the curved surface area of a hemisphere is $2 \pi r^2$ .
$CSA_{hemi} = 2 \times \frac{22}{7} \times (7 \text{ cm})^2$ $CSA_{hemi} = 2 \times \frac{22}{7} \times 49 \text{ cm}^2$ $CSA_{hemi} = 2 \times 22 \times 7 \text{ cm}^2$ $CSA_{hemi} = 308 \text{ cm}^2$
Calculate the total surface area of the structure.
The total surface area is the sum of the curved surface area of the cone and the curved surface area of the hemisphere (the base of the cone is covered by the hemisphere).
$\text{Total Surface Area} = CSA_{cone} + CSA_{hemi}$ $\text{Total Surface Area} = 550 \text{ cm}^2 + 308 \text{ cm}^2$ $\text{Total Surface Area} = 858 \text{ cm}^2$
Express the answer correct to 3 significant figures.
$858 \text{ cm}^2$ is already expressed to 3 significant figures.
The surface area of the structure is $\boxed{858 \text{ cm}^2}$ .

Define the following terms: (i) measurement (ii) quantity (iii) unit (iv) accuracy (v) precision

Step-by-step solution

GSE 212: BASIC GENERAL MATHEMATICS III (1C)

Express $x$ in terms of $y$ .

Find $x$ when $y = 5$ .

Find $x$ when $y = 10$ .

Find the length of the base PR using the area of triangle PSR.

Determine the length of PQ.

Calculate the value of $|QR|$.

Multiply both sides by 21.

Divide both sides by $4b^2$ .

Add $\frac{3}{5}b$ to both sides.

Substitute the given values into the formula for $d$ .

Perform the calculations.

Round the answer to the nearest 5m.

Calculate the slant height ( $l$ ) of the cone.

Calculate the curved surface area of the cone ( $CSA_{cone}$ ).

Calculate the curved surface area of the hemisphere ( $CSA_{hemi}$ ).

Calculate the total surface area of the structure.

Express the answer correct to 3 significant figures.

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More Mathematics Questions

Define the following terms: (i) measurement (ii) quantity (iii) unit (iv) accuracy (v) precision

Step-by-step solution

GSE 212: BASIC GENERAL MATHEMATICS III (1C)

Express xxx in terms of yyy.

Find xxx when y=5y = 5y=5.

Find xxx when y=10y = 10y=10.

Find the length of the base PR using the area of triangle PSR.

Determine the length of PQ.

Calculate the value of $|QR|$.

Multiply both sides by 21.

Divide both sides by 4b24b^24b2.

Add 35b\frac{3}{5}b53​b to both sides.

Substitute the given values into the formula for ddd.

Perform the calculations.

Round the answer to the nearest 5m.

Calculate the slant height (lll) of the cone.

Calculate the curved surface area of the cone (CSAconeCSA_{cone}CSAcone​).

Calculate the curved surface area of the hemisphere (CSAhemiCSA_{hemi}CSAhemi​).

Calculate the total surface area of the structure.

Express the answer correct to 3 significant figures.

Still stuck on this one?

More Mathematics Questions

Express $x$ in terms of $y$ .

Find $x$ when $y = 5$ .

Find $x$ when $y = 10$ .

Divide both sides by $4b^2$ .

Add $\frac{3}{5}b$ to both sides.

Substitute the given values into the formula for $d$ .

Calculate the slant height ( $l$ ) of the cone.

Calculate the curved surface area of the cone ( $CSA_{cone}$ ).

Calculate the curved surface area of the hemisphere ( $CSA_{hemi}$ ).