Question

kaj is flying a kite, holding her hands a distance of 3.5 feet above the ground and letting all the kite’s string out. she measures the angle of elevation from her hand to the kite to be 29°. if the string from the kite to her hand is 130 feet long, how many feet is the kite above the ground? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We can use the sine function in a right triangle, where the string length is the hypotenuse, and the height from the hand to the kite is the opposite side to the angle of elevation. The formula for sine is $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 29^\circ$, hypotenuse $= 130$ feet, and the opposite side (height from hand to kite) is $x$. So, $\sin(29^\circ)=\frac{x}{130}$.

Step2: Solve for $x$

Multiply both sides by 130 to get $x = 130\times\sin(29^\circ)$. Calculate $\sin(29^\circ)\approx0.4848$, then $x\approx130\times0.4848 = 63.024$ feet.

Step3: Add the height of the hand above the ground

The hand is 3.5 feet above the ground, so the total height of the kite above the ground is $63.024 + 3.5 = 66.524$ feet. Rounding to the nearest hundredth, we get $66.52$ feet (wait, actually, let's recalculate more accurately. $\sin(29^\circ)$ is approximately 0.4848096203. So $130\times0.4848096203 = 63.02525064$. Then add 3.5: $63.02525064 + 3.5 = 66.52525064$, which rounds to 66.53 when rounded to the nearest hundredth? Wait, no, 66.52525... the third decimal is 5, so we round up the second decimal. So 66.53? Wait, let's check the calculation again.

Wait, $\sin(29^\circ)$: using a calculator, $\sin(29^\circ) \approx 0.48480962024633706$. Then $130 * 0.48480962024633706 = 130 * 0.48480962024633706 = 63.02525063202382$. Then add 3.5: $63.02525063202382 + 3.5 = 66.52525063202382$. Now, rounding to the nearest hundredth: the number is 66.52525..., so the hundredth place is 2, the thousandth is 5, so we round up the hundredth place: 66.53? Wait, no, 66.52525... the decimal is 66.52 (tenths and hundredths) and then 5 thousandths. So 66.525, so when rounding to the nearest hundredth, look at the third decimal: 5, so we add 1 to the second decimal. So 66.53? Wait, no, 66.525 is equal to 66.53 when rounded to the nearest hundredth? Wait, 66.525: the hundredth digit is 2, the next digit is 5, so we round up the 2 to 3, so 66.53. Wait, but let's check with more precise calculation.

Wait, maybe I made a mistake in the initial step. Let's recalculate:

$\sin(29^\circ) \approx 0.48480962024633706$

$130 * 0.48480962024633706 = 63.02525063202382$

Add 3.5: $63.02525063202382 + 3.5 = 66.52525063202382$

Now, to round to the nearest hundredth, we look at the thousandth place, which is 5. So we round up the hundredth place: 2 becomes 3. So 66.53? Wait, but 66.525 is halfway between 66.52 and 66.53, but in rounding, 5 rounds up. So yes, 66.53. Wait, but maybe my initial multiplication was wrong. Let's check 130 * 0.48480962024633706:

130 * 0.48480962024633706:

0.48480962024633706 * 100 = 48.480962024633706

0.48480962024633706 * 30 = 14.544288607390112

Add them: 48.480962024633706 + 14.544288607390112 = 63.02525063202382. Correct. Then add 3.5: 66.52525063202382. So rounding to the nearest hundredth: the second decimal is 2, third is 5, so round up to 66.53. Wait, but maybe the problem expects two decimal places. Let's confirm.

Wait, maybe I miscalculated the sine. Let's use a calculator for $\sin(29^\circ)$:

Using a calculator, $\sin(29^\circ) \approx 0.48480962024633706$. So 130 * that is 63.02525063202382. Add 3.5: 66.52525063202382. So to the nearest hundredth, that's 66.53? Wait, no, 66.52525... the hundredth place is 2, the thousandth is 5, so we round up the hundredth place: 2 becomes 3, so 66.53. Wait, but maybe the initial calculation of the sine was wrong? No, that's the correct value. So the total height is approximately 66.53 feet? Wait, but let's check again.

Wait, maybe the problem is that when I did 63.02525 + 3.5, that's 66.52525, which is 66.53 when rounded to the nearest hundredth (because the third decimal is 5, so we round the second decimal up from 2 to 3). Yes.

Answer:

66.53