Sydney drew a scale diagram of a circular fire pit in the centre of a circular patio with an actual circumference of 15m. The circle containing the fire put is a reduction of the circular patio by a scale factor of1/3.
A.) Determine the diameter of the actual fire pit to the nearest hundredth of a metre.
B.) What is the diameter of the fire pit in the scale diagram if the scale factor used was 2/15?
C.) After construction, the raised fire pit will have a surface area of 6.47m^3. What will be the surface area of a similar fire pit that is enlarged by a scale factor of 3?
D.) After construction, the raised fire put will have a volume of0.99m^3. If the volume of the enlargement is 26.73m^3, what scale factor was used?
To solve these questions, we will use the information provided and apply the appropriate formulas. Let's go step by step:
A) To determine the diameter of the actual fire pit, we need to find the diameter of the circular patio. We know that the circumference of the patio is 15m.
We know that the formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter of the circle.
Therefore, we can rearrange the formula to solve for the diameter:
d = C/π
Substituting the given value of the circumference, we get:
d = 15/π ≈ 4.77m
So, the diameter of the actual fire pit is approximately 4.77m.
B) The diameter of the fire pit in the scale diagram can be determined by multiplying the scale factor by the diameter of the actual fire pit. The scale factor provided is 2/15.
Let's first find the diameter of the fire pit in the scale diagram:
diameter in scale diagram = (2/15) * 4.77m
Calculating this, we get:
diameter in scale diagram ≈ 0.636m
Therefore, the diameter of the fire pit in the scale diagram is approximately 0.636m.
C) To find the surface area of the enlarged fire pit, we need to use the concept that similar figures have a scale factor squared relationship for their areas. Since the fire pit is being enlarged by a scale factor of 3, the surface area of the enlarged fire pit will be (3^2 = 9) times the surface area of the original fire pit.
Given that the surface area of the original fire pit is 6.47m^2, we can calculate the surface area of the enlarged fire pit as follows:
surface area of enlarged fire pit = 9 * 6.47m^2
Calculating this, we get:
surface area of enlarged fire pit ≈ 58.23m^2
Therefore, the surface area of the enlarged fire pit will be approximately 58.23m^2.
D) To determine the scale factor used based on the volumes of the fire pit, we need to use the concept that similar figures have a scale factor cubed relationship for their volumes.
Given that the volume of the actual fire pit is 0.99m^3 and the volume of the enlarged fire pit is 26.73m^3, we can set up the following equation:
(3^3) * 0.99m^3 = 26.73m^3
Simplifying this equation, we get:
27 * 0.99m^3 = 26.73m^3
Solving for the scale factor, we get:
Scale factor = (26.73m^3) / (27 * 0.99m^3) ≈ 1
Therefore, the scale factor used for the enlargement is approximately 1.
A.) To determine the diameter of the actual fire pit, we know that the circumference of the circular patio is 15m. The formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter. We can rearrange this formula to solve for the diameter: d = C/π. Plugging in the given value for the circumference, we get d = 15m/π ≈ 4.77m.
B.) If the scale factor used in the scale diagram is 2/15, we can determine the diameter of the fire pit in the scale diagram by multiplying the actual diameter by the scale factor. Given the actual diameter from the previous calculation is approximately 4.77m, we can multiply it by 2/15: (4.77m)(2/15) ≈ 0.636m.
C.) If the raised fire pit has a surface area of 6.47m^2 and is being enlarged by a scale factor of 3, we can determine the surface area of the enlarged fire pit by squaring the scale factor. The formula for the surface area of a circle is A = πr^2, where A is the surface area and r is the radius. Since the scale factor applies to both the radius and the area, we can calculate the surface area of the enlarged fire pit as follows: (6.47m^2)(3^2) = 58.23m^2.
D.) If the volume of the raised fire pit is 0.99m^3 and the volume of the enlarged fire pit is 26.73m^3, we can determine the scale factor by taking the cube root of the ratio of the volumes. The formula for the volume of a sphere is V = (4/3)πr^3. Since the scale factor applies to the radius and the volume, we can calculate the scale factor as follows: (26.73m^3/0.99m^3)^(1/3) ≈ 1.95.
Therefore:
A.) The diameter of the actual fire pit is approximately 4.77m.
B.) The diameter of the fire pit in the scale diagram is approximately 0.636m.
C.) The surface area of the enlarged fire pit is approximately 58.23m^2.
D.) The scale factor used for the enlargement is approximately 1.95.
Since the actual diameter of the fire pit is 15m and the formula for the circumference is C = πd, substitute the circumference of 15m into the formula and solve for d.
15 = πd and then divide both sides by π getting d = 15/π = 5 rounded to the nearest meter. The scale drawing is smaller so (2/15) times 5 is 10/15 or 2/3 m
If the raised fire pit has a surface area of 6.47 and a similar one larger by a factor of 3 would have a surface area of
6.47 times 3 = 19.41 or 20m2 rounded to the nearest square meter.
For the last part, if the volume of the fire pit is .99m3 and the enlargement is 26.73m3, the scale factor used is
.99x = 26.73 ror x = 27