Rewrite the expression as a single fraction: sec(x) + tan(x)
Using trigonometric identities and reciprocals, we can rewrite the above expression in terms of sin(x) and cos(x) to be better able to simplify things. Doing that, we have: sec(x) + tan(x) = 1/cos(x) + sin(x)/cos(x). Since we have a common denominator in cos(x), we can simply add the numerators to make this a single fraction: (1 + sin(x)) / cos(x).
Air is being pumped into a spherical balloon at the rate of 4.5 cubic inches per minute. Find the rate of change of the radius when the radius is exactly 2 inches.
Since we want to find out how the radius of a balloon is changing at the same time the volume of the balloon is changing, we start with the formula for volume of a sphere which is V = (4/3)PI(R^3) where R is the radius and PI is the irrational number approximately 3.14. To find out how the volume is changing, we differentiate the equation implicitly with respect to t (time). So we get dV/dt = 4PIR^2(dR/dt). Now we can fill in the given pieces of the equation. We know that the volume is changing at a rate of 4.5 cubic inches per minute, so dV/dt = 4.5. We also know that we want to find the radius' rate of change when the radius exactly equals 2 inches, so R = 2. Plugging these knowns into the equation yields 4.5 = 4PI(4)(dR/dt), which simplifies to 4.5 = 16PI(dR/dt). We want to find dR/dt, so we can divide both sides of the equation by 16PI and see that our answer is 0.09 cubic inches per minute. This is the rate at which the radius is changing as we blow air into the balloon.
The wood material used for the roof of an ancient Japanese temple comes from northern Europe. The wooden roof must withstand as much as 100cm of snow in the winter. Architects conducted a study to estimate the mean bending strength of the wood material used for the roof. A sample of 25 pieces of wood were tested and yielded the following estimates of breaking strength: mean = 75.4cm, standard deviation = 10.9cm. Estimate the true average breaking strength of the wood using a 90% confidence interval and interpret the result.
Using the formula for confidence intervals, we see that the formula translates to (75.4 - ((1.711)(10.9/5)) and (75.4 + ((1.711)(10.9/5)). This means our confidence interval for the true average breaking strength is (71.67, 79.13). This presents a difficult situation for the architects in that the wood must be able to sufficiently hold up to 100cm of snow, however according to our calculations, 90% of the time, the wood will only hold between 71.67cm and 79.13cm of snow. Therefore we can conclude that the wood used for the roof is not strong enough to withstand the snowfall amounts of a Japanese winter.