A shadow moves around a sundial 15^(∘) every hour.
We need to find out how many hours need to pass for the angle of rotation of the shadow to equal 8π5 radians. First, let's rewrite the radian measure in degrees by multiplying it by a conversion factor 180^(∘)π radians. Next, we will simplify.
Now that we have converted the measure, we can focus on finding the time it takes for the shadow to rotate 288^(∘). Since we know that one hour corresponds to an angle of 15^(∘), we can write and solve a proportion for the duration t.
1 h/15^(∘) = t/288^(∘) ⇒ t= 19.2 h
It will take 19.2 hours, which is 19 hours and 12 minutes, for the shadow to complete the turn.
b
We need to find the angle of rotation of the shadow (in radians) after 5 hours. Here is our plan.
Find the degree measure of the angle.
Convert the degree measure of the angle to radians. To begin, notice that an angle of rotation x is always proportional to the time that passed. We can use this fact to find its measure.
1 h/15^(∘) = 5 h/x ⇒ x= 75^(∘)
We found that the shadow turned 75 degrees, which we will now rewrite in radian measure. To do that, we need to multiply the measure by a conversion factor of π radians180^(∘) and simplify.
A sundial has a radius of 8 inches. We need to find the length of an arc formed by a shadow after 14 hours. First, we are going to find the degree measure of the angle x. To do that, we will use a proportion.
1 h/15^(∘) = 14 h/x ⇒ x= 210^(∘)
Let's now recall the formula for the arc length s. In the formula r is the radius and θ is the measure of the central angle, expressed in radians.
s=rθ
Since we only have the angle measure expressed in degrees, we need to convert it to radians. Once again, we will multiply the number of degrees by a conversion factor and then simplify.
