Question

question #11 reference q.14159 on a particular day, the eiffel tower in paris casts a shadow of 599 m. use the sketch to determine the height of the tower. to the nearest meter, the height of the tower is question #12 reference q.14160 in right triangle abc, angle abc = 90° angle bac = 70° and ac = 29 units. to the nearest whole number, the perimeter of the triangle is ____.

Explanation:

Step1: Identify the trigonometric relationship for Question #11

We have a right - triangle where the length of the shadow is the adjacent side to the given angle and the height of the tower is the opposite side. We use the tangent function: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let $\theta = 28.4^{\circ}$ and the adjacent side $x = 599$ m. So, $\tan(28.4^{\circ})=\frac{h}{599}$, where $h$ is the height of the tower.

Step2: Solve for the height of the tower in Question #11

$h = 599\times\tan(28.4^{\circ})$. Using a calculator, $\tan(28.4^{\circ})\approx0.542$. Then $h=599\times0.542 = 599\times\frac{542}{1000}=324.658\approx325$ m.

Step3: Identify trigonometric relationships for Question #12

In right - triangle $ABC$ with $\angle ABC = 90^{\circ}$, $\angle BAC=70^{\circ}$ and $AC = 29$ units. First, find $AB$ (adjacent to $\angle BAC$) using $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, so $\cos(70^{\circ})=\frac{AB}{29}$, then $AB = 29\times\cos(70^{\circ})$. And find $BC$ (opposite to $\angle BAC$) using $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, so $\sin(70^{\circ})=\frac{BC}{29}$, then $BC = 29\times\sin(70^{\circ})$.

Step4: Calculate $AB$ and $BC$ in Question #12

$\cos(70^{\circ})\approx0.342$, so $AB = 29\times0.342 = 9.918$. $\sin(70^{\circ})\approx0.940$, so $BC = 29\times0.940 = 27.26$.

Step5: Calculate the perimeter of the triangle in Question #12

The perimeter $P=AB + BC+AC$. Substitute the values: $P=9.918 + 27.26+29=66.178\approx66$ units.

Answer:

Question #11: 325 m
Question #12: 66 units