Example Question - calculating length
Here are examples of questions we've helped users solve.
Calculating Length of AD in Trapezoidal Prism
To solve for the length of \( AD \) in the trapezoidal prism, we need to use the Pythagorean Theorem in two steps because we are dealing with a three-dimensional figure. First, we can find the length of \( BD \) by applying the Pythagorean Theorem to triangle \( BCD \), which is a right-angled triangle: \[ BD = \sqrt{BC^2 + CD^2} \] \[ BD = \sqrt{12^2 + 6^2} \] \[ BD = \sqrt{144 + 36} \] \[ BD = \sqrt{180} \] \[ BD = 13.4164 \text{ cm} \] (approximated to four decimal places) Now we can find the length of \( AD \) by using the Pythagorean Theorem again, this time for the right-angled triangle \( ABD \), where \( AB \) is the base, \( BD \) is the height, and \( AD \) is the hypotenuse: \[ AD = \sqrt{AB^2 + BD^2} \] \[ AD = \sqrt{10^2 + 13.4164^2} \] \[ AD = \sqrt{100 + 179.9992} \] \[ AD = \sqrt{279.9992} \] \[ AD = 16.7426 \text{ cm} \] (approximated to four decimal places) It seems there might be an error in the question or in the calculation, as based on the given dimensions, \( AD \) computes to 16.7426 cm rather than the 12.649 cm stated in the question. Please double-check the dimensions provided in the question and ensure any additional details or constraints need to be considered. If the dimensions given are correct, the calculated length of \( AD \) will be approximately 16.7426 cm instead of 12.649 cm.
Calculating Length in a Right Triangle
To find the length of \(x\) in the right triangle as depicted in the image, we can use trigonometric ratios. The triangle has an angle of 35 degrees, and we're given the length of the side opposite to this angle, which is 16 units. Since we have the angle and the opposite side, we can use the tangent function (tan) to find the length of the adjacent side (\(x\)). The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. So, we have: \[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \] Substituting the known values, we get: \[ \tan(35^\circ) = \frac{16}{x} \] To isolate \(x\), we'll multiply both sides by \(x\) and then divide both sides by \(\tan(35^\circ)\): \[ x \cdot \tan(35^\circ) = 16 \] \[ x = \frac{16}{\tan(35^\circ)} \] Using a calculator and making sure it's set to degree mode, we can calculate the value of \(x\): \[ x \approx \frac{16}{0.70020753820971} \] \[ x \approx 22.849 \] So, the length of \(x\) rounded to three decimal places is approximately 22.849 units.
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