the equatorial radius of the earth is approximately 3960 miles. suppose a wire is wrapped tightly around the earth at the equator. how much must this wire be lengthened if it is to be strung on 10 feet poles above the ground? (1 mile=5280 feet)
delta y= dy/dx (2pir) * deltax
delta y= 2pi * 10/5280
delta y= 20pi/5280
is this right or no? thanks
No Calculus needed for this
circumference of earth = 2π(radius) = 2πr
radius of earth with string around it = r+10
circumference with string = 2π(r+10)
= 2πr + 20π
so the increase in the length of string
= 20π ft or appr 62.8 ft
notice the data given was not even necessary.
No, the calculation is not correct. To find the length that the wire must be lengthened, we need to calculate the change in circumference of the equator.
The formula to calculate the circumference of a circle is given by:
C = 2πr
Given that the equatorial radius of the Earth is approximately 3960 miles, the initial circumference can be calculated as follows:
C_initial = 2π * 3960 = 24,840 miles
To find the change in circumference when the wire is strung on 10-foot poles above the ground, we first need to convert 10 feet to miles:
10 feet = 10/5280 miles
Now, we can calculate the change in circumference as follows:
Change in circumference = 2π * (3960 + 10/5280) - 2π * 3960
Simplifying the equation:
Change in circumference = 2π * (3960 + 10/5280 - 3960)
Change in circumference = 2π * (3960 + 10/5280 - 3960)
Approximately, the change in circumference will be 0.001before the units of the radius were changed back to miles.
Therefore, the wire must be lengthened by approximately 0.001 miles (or 5.28 feet) to be strung on 10-foot poles above the ground.
To determine how much the wire must be lengthened to be strung on 10 feet poles above the ground, we can use the formula for the circumference of a circle:
C = 2πr
where C is the circumference, π is approximately 3.14159, and r is the radius.
In this case, the wire is wrapped tightly around the Earth at the equator, so the circumference of the wire is equal to the equatorial circumference of the Earth.
Given that the equatorial radius of the Earth (r) is approximately 3960 miles, we can calculate the equatorial circumference (C) as follows:
C = 2π * 3960
C ≈ 2π * 3960 ≈ 2 * 3.14159 * 3960 ≈ 24881.53 miles
Now that we have the circumference, we need to calculate how much the wire must be lengthened to be strung on 10 feet poles above the ground.
Since 1 mile is equal to 5280 feet, we can convert the circumference of the wire from miles to feet:
C_feet = 24881.53 miles * 5280 feet/mile
C_feet ≈ 131,480,018.4 feet
Next, we need to find the difference in length between the circumference of the wire and the required height of the poles, which is 10 feet:
Difference = C_feet - 10 feet
Difference ≈ 131,480,018.4 feet - 10 feet ≈ 131,480,008.4 feet
Therefore, the wire must be lengthened by approximately 131,480,008.4 feet if it is to be strung on 10 feet poles above the ground.
Your initial calculation is not correct. You used the equation for the differential change in height, but you need to calculate the actual change in length of the wire. Additionally, you need to convert the units correctly and take the full circumference of the Earth into account, not just a small interval.