For each object, its surface area, and some dimensions are given. Calculate the dimension indicated by the variable to the nearest tenth of a unit.
a) right cone
SA=7012mm^2
GIVEN=diameter of 48mm
b) right square pyramid
SA=65.5m^2
GIVEN=length of 3.5m
To find the dimension indicated by the variable, we need to use the surface area formula of the respective object and solve for the variable.
a) Right Cone:
The surface area (SA) of a right cone is given by the formula:
SA = πr(r + l)
Given: Diameter (d) = 48mm
To find the radius (r) from the diameter, we divide it by 2:
r = d/2 = 48mm/2 = 24mm
Now, we can substitute the values in the surface area formula and solve for the variable (l):
7012mm^2 = π(24mm)(24mm + l)
Divide both sides of the equation by (π * 24mm) to isolate (24mm + l):
7012mm^2 / (π * 24mm) = 24mm + l
To find l, subtract 24mm from both sides:
7012mm^2 / (π * 24mm) - 24mm = l
Use a calculator to evaluate the right-hand side of the equation, which will give you the value of l in millimeters. Round the result to the nearest tenth of a unit.
b) Right Square Pyramid:
The surface area (SA) of a right square pyramid is given by the formula:
SA = s^2 + 2s(l)
Given: Length (s) = 3.5m
To find the slant height (l) from the surface area and length, we can rearrange the formula:
65.5m^2 = s^2 + 2s(l)
Rearranging the equation gives:
2s(l) = 65.5m^2 - s^2
Divide both sides by 2s to isolate l:
l = (65.5m^2 - s^2) / (2s)
Substitute the given value of s into the equation and evaluate the right-hand side. Round the result to the nearest tenth of a unit to find the value of l in meters.