We want to find the and the of the following hat.
Let's start by finding the slant height. To do so, we can use the fact the slant height of a cone makes a with the height and the .
Notice that we know the lengths of the of the triangle. To find the , we will use the .
a2+b2=c2
In the formula,
a and
b are the lengths of the legs and
c is the length of the hypotenuse of a right triangle. We are given the triangle with
a=7 and
b=14. Let's apply the Pythagorean Theorem to this triangle.
a2+b2=c2⇕72+142=c2
Now we can solve an that we got to find
c.
72+142=c2
49+196=c2
245=c2
15.652475…=c
c=15.652…
c≈15.7
The slant height of the hat is about
15.7 inches. Next, we will find the lateral area of the hat. Recall that the lateral area of a cone is equal to one-half the of the base times the slant height. To calculate the lateral area of a cone, we can use the following formula.
L.A.=πrℓ
In this formula,
r is the radius of the base and
ℓ is the slant height of the cone. We can substitute the slant height and the radius of the conical hat into the formula and calculate its lateral area. Let's do it!
L.A.=πrℓ
L.A.=π(7)(15.7)
L.A.=109.9π
L.A.=345.261032…
L.A.≈345.3
We got that the lateral area of the hat is about
345.3 square inches.