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### Question 58:
A cone has a volume of [tex]\(1256 \, \text{cm}^3\)[/tex], and the height [tex]\(h\)[/tex] of the cone is 14 cm. Find the radius of the cone. The value of [tex]\(\pi\)[/tex] is 3.142.
Given:
- Volume [tex]\(V = 1256 \, \text{cm}^3\)[/tex]
- Height [tex]\(h = 14 \, \text{cm}\)[/tex]
- [tex]\(\pi = 3.142\)[/tex]
The formula for the volume of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Rearrange the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Substitute the given values into the formula:
[tex]\[ r^2 = \frac{3 \times 1256}{3.142 \times 14} \][/tex]
Let's denote the radius squared as [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = 85.65972537964899 \][/tex]
Taking the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{85.65972537964899} \approx 9.255253933828557 \, \text{cm} \][/tex]
Among the given choices, the closest value to [tex]\(9.255253933828557 \, \text{cm}\)[/tex] is:
B. [tex]\(9.3 \, \text{cm}\)[/tex]
### Question 59:
Solve [tex]\(\frac{2}{3} - \frac{1}{y} = \frac{5}{y}\)[/tex]
First, combine the terms involving [tex]\(y\)[/tex] on one side of the equation:
[tex]\[ \frac{2}{3} = \frac{5}{y} + \frac{1}{y} \][/tex]
Combine the fractions on the right side:
[tex]\[ \frac{2}{3} = \frac{5+1}{y} \][/tex]
[tex]\[ \frac{2}{3} = \frac{6}{y} \][/tex]
To clear the fraction, multiply both sides by [tex]\(y\)[/tex]:
[tex]\[ y \cdot \frac{2}{3} = 6 \][/tex]
Multiply both sides by [tex]\(3\)[/tex] to clear the denominator:
[tex]\[ 2y = 18 \][/tex]
Divide both sides by [tex]\(2\)[/tex]:
[tex]\[ y = 9 \][/tex]
So, the solution is:
D. [tex]\(9\)[/tex]
### Question 60:
Change [tex]\(\frac{23}{7}\)[/tex] to a mixed fraction.
First, divide [tex]\(23\)[/tex] by [tex]\(7\)[/tex]:
[tex]\[ 23 \div 7 = 3 \text{ remainder } 2 \][/tex]
So, [tex]\(\frac{23}{7}\)[/tex] can be written as:
[tex]\[ 3 \frac{2}{7} \][/tex]
The answer is:
A. [tex]\(3 \frac{2}{7}\)[/tex]
In summary, the answers are:
- Q58: B. [tex]\( \quad 9.3 \, \text{cm} \)[/tex]
- Q59: D. [tex]\( 9 \)[/tex]
- Q60: A. [tex]\( 3 \frac{2}{7} \)[/tex]
### Question 58:
A cone has a volume of [tex]\(1256 \, \text{cm}^3\)[/tex], and the height [tex]\(h\)[/tex] of the cone is 14 cm. Find the radius of the cone. The value of [tex]\(\pi\)[/tex] is 3.142.
Given:
- Volume [tex]\(V = 1256 \, \text{cm}^3\)[/tex]
- Height [tex]\(h = 14 \, \text{cm}\)[/tex]
- [tex]\(\pi = 3.142\)[/tex]
The formula for the volume of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Rearrange the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Substitute the given values into the formula:
[tex]\[ r^2 = \frac{3 \times 1256}{3.142 \times 14} \][/tex]
Let's denote the radius squared as [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = 85.65972537964899 \][/tex]
Taking the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{85.65972537964899} \approx 9.255253933828557 \, \text{cm} \][/tex]
Among the given choices, the closest value to [tex]\(9.255253933828557 \, \text{cm}\)[/tex] is:
B. [tex]\(9.3 \, \text{cm}\)[/tex]
### Question 59:
Solve [tex]\(\frac{2}{3} - \frac{1}{y} = \frac{5}{y}\)[/tex]
First, combine the terms involving [tex]\(y\)[/tex] on one side of the equation:
[tex]\[ \frac{2}{3} = \frac{5}{y} + \frac{1}{y} \][/tex]
Combine the fractions on the right side:
[tex]\[ \frac{2}{3} = \frac{5+1}{y} \][/tex]
[tex]\[ \frac{2}{3} = \frac{6}{y} \][/tex]
To clear the fraction, multiply both sides by [tex]\(y\)[/tex]:
[tex]\[ y \cdot \frac{2}{3} = 6 \][/tex]
Multiply both sides by [tex]\(3\)[/tex] to clear the denominator:
[tex]\[ 2y = 18 \][/tex]
Divide both sides by [tex]\(2\)[/tex]:
[tex]\[ y = 9 \][/tex]
So, the solution is:
D. [tex]\(9\)[/tex]
### Question 60:
Change [tex]\(\frac{23}{7}\)[/tex] to a mixed fraction.
First, divide [tex]\(23\)[/tex] by [tex]\(7\)[/tex]:
[tex]\[ 23 \div 7 = 3 \text{ remainder } 2 \][/tex]
So, [tex]\(\frac{23}{7}\)[/tex] can be written as:
[tex]\[ 3 \frac{2}{7} \][/tex]
The answer is:
A. [tex]\(3 \frac{2}{7}\)[/tex]
In summary, the answers are:
- Q58: B. [tex]\( \quad 9.3 \, \text{cm} \)[/tex]
- Q59: D. [tex]\( 9 \)[/tex]
- Q60: A. [tex]\( 3 \frac{2}{7} \)[/tex]
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