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Similar Question 1
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p>Modify your answers to parts a) and b) to relate the total surface area, <code class='latex inline'>S</code>, and the side length, <code class='latex inline'>l</code>. </p>
Similar Question 2
<p>Find the volume of a cube with surface area of </p> <ul> <li>i) <code class='latex inline'>294 m^2</code></li> <li>ii) <code class='latex inline'>36.8 m^2</code></li> </ul>
Similar Question 3
<p>Solve. </p><p><code class='latex inline'> \displaystyle \left( \frac{1}{16} \right)^{\frac{1}{4}} - \sqrt[3]{\frac{8}{27}} = \sqrt{x^2} </code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p> Write a formula to express the side length, <code class='latex inline'>l</code>, in terms of the area, <code class='latex inline'>A</code>, using a rational exponent. </p>
<p>Evaluate. </p><p><code class='latex inline'> \displaystyle 27^{\frac{2}{3}} - 81^{\frac{3}{4}} </code></p>
<p> Simplify the result to express the surface area of a cylinder with a height of 10 m in terms of its volume.</p>
<p>State whether each expression is true or false.</p><p><code class='latex inline'> \displaystyle \left(\frac{1}{a} \times \frac{1}{b} \right)^{-1} = ab </code></p>
<p>The formulas <code class='latex inline'>h = 241m^{\frac{1}{4}}</code> and <code class='latex inline'>r = \frac{107}{2}m^{-\frac{1}{4}}</code> give the heartbeat frequency, <code class='latex inline'>h</code>, in beats per minute, and respiratory frequency, <code class='latex inline'>r</code>, in breaths per minute, for a resting animal with mass <code class='latex inline'>m</code>, in kilograms. </p><p>Use the formula <code class='latex inline'>B = \frac{1}{100}m^{\frac{2}{3}}</code> to determine the brain mass, <code class='latex inline'>B</code>, for </p> <ul> <li>i) killer whale: 6400 kg</li> <li>ii) dog 6.4 kg.</li> <li>iii) mous: 0.064 kg.</li> </ul>
<p>If <code class='latex inline'>\displaystyle M = \frac{(16x^{8}y^{-4})^\frac{1}{4}}{32x^{-2}y^{8}}</code>, determine values for <code class='latex inline'>x</code> and <code class='latex inline'>y</code> so that <code class='latex inline'>M > 1</code></p>
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p> Write a formula to express the area, <code class='latex inline'>A</code>, of one face in terms of the side length, <code class='latex inline'>l</code>.</p>
<p>The volume and surface area of a cylinder are given, respectively, by the formulas</p><p><code class='latex inline'>V = \pi r^2 h</code> and <code class='latex inline'>SA = 2\pi rh + 2\pi r^2</code>.</p><p>Determine an expression, in simplified form, that represents the surface area-to-volume ratio for a cylinder.</p>
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p>Modify your answers to parts a) and b) to relate the total surface area, <code class='latex inline'>S</code>, and the side length, <code class='latex inline'>l</code>. </p>
<p>Solve. </p><p><code class='latex inline'> \displaystyle \left( \frac{1}{16} \right)^{\frac{1}{4}} - \sqrt[3]{\frac{8}{27}} = \sqrt{x^2} </code></p>
<p>Evaluate. </p><p><code class='latex inline'> \displaystyle 16^{\frac{3}{4}} + 16^{\frac{3}{4}} - 81^{-\frac{1}{4}} </code></p>
<p>Solve. </p><p><code class='latex inline'> \displaystyle \sqrt[3]{\frac{1}{8}} - \sqrt[4]{x^4} + 15 = \sqrt[4]{16} </code></p>
<p>Evaluate. </p><p><code class='latex inline'> \displaystyle 16^{\frac{3}{2}} + 16^{-0.5} + 8 - 27^{\frac{2}{3}} </code></p>
<p>If <code class='latex inline'>M = \frac{(16x^{8}y^{-4})^\frac{1}{4}}{32x^{-2}y^{8}}</code>, determine values for <code class='latex inline'>x</code> and <code class='latex inline'>y</code> so that <code class='latex inline'>0 < M < 1</code></p>
<p>State whether each expression is true or false.</p><p><code class='latex inline'> \displaystyle \left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)^6 = x^2 + y^2 </code></p>
<p>The formulas <code class='latex inline'>h = 241m^{\frac{1}{4}}</code> and <code class='latex inline'>r = \frac{107}{2}m^{-\frac{1}{4}}</code> give the heartbeat frequency, <code class='latex inline'>h</code>, in beats per minute, and respiratory frequency, <code class='latex inline'>r</code>, in breaths per minute, for a resting animal with mass <code class='latex inline'>m</code>, in kilograms. </p><p>Describe what happens to each frequency as the mass of the animal decreases.</p>
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p> What is the side length of a cube for which each square face has an area of </p> <ul> <li><strong>i)</strong> <code class='latex inline'>36 m^2</code>?<br></li> <li><strong>ii)</strong> <code class='latex inline'>169 cm^2</code>? </li> <li><strong>iii)</strong> <code class='latex inline'>80 m^2</code>? </li> </ul>
<p>Write a formula to express the volume of the cube, <code class='latex inline'>V</code>, in terms of the side length, <code class='latex inline'>l</code>.</p>
<p>The formulas <code class='latex inline'>h = 241m^{\frac{1}{4}}</code> and <code class='latex inline'>r = \frac{107}{2}m^{-\frac{1}{4}}</code> give the heartbeat frequency, <code class='latex inline'>h</code>, in beats per minute, and respiratory frequency, <code class='latex inline'>r</code>, in breaths per minute, for a resting animal with mass <code class='latex inline'>m</code>, in kilograms. </p><p>Determine the heat beat frequency and respiratory frequency for each animal.</p> <ul> <li>i) killer whale: 6400 kg</li> <li>ii) dog 6.4 kg.</li> <li>iii) mous: 0.064 kg.</li> </ul>
<p>The volume of a cube is 0.015626 <code class='latex inline'>m^3</code>. Determine the length of each side.</p>
<p>Write a formula to express the side length, <code class='latex inline'>l</code>, in terms of the volume, <code class='latex inline'>V</code>, using a rational exponent. </p>
<p>Explain why <code class='latex inline'>(-100)^{0.2}</code> is possible to evaluate while <code class='latex inline'>(-100)^{0.5}</code> is not.</p>
<p>Find the surface area of a cube with a volume of cub is</p> <ul> <li>i) <code class='latex inline'>1000 m^3</code></li> <li>ii) <code class='latex inline'>200 m^3</code></li> </ul>
<p>What is the square-cube law? consider the following sequence of cubes.</p><img src="/qimages/813" /><p>Use the fact that <code class='latex inline'>l = \sqrt{\dfrac{SA}{6}}</code> to calculate the side length of a cube with a surface area of</p> <ul> <li><strong>i)</strong> <code class='latex inline'>150 m^2</code><br></li> <li><strong>ii)</strong> <code class='latex inline'>600 cm^2</code> </li> <li><strong>iii)</strong> <code class='latex inline'>250 m^2 </code></li> </ul>
<p>(a) What is the formula for the volume of a cylinder?</p><p>(b) Rewrite this formula to express the radius as a function of the volume and height of the cylinder.</p>
<p>The surface area, S, of a sphere can be expressed in terms of its volume, V, using the formula <code class='latex inline'>S(V) = (4\pi)^{\frac{1}{3}}(3V)^{\frac{2}{3}}</code>. A beach ball has volume <code class='latex inline'>24 000 cm^3</code>. Find its surface area, to the nearest hundred square centimetres. </p>
<p>Write the Surface Area as a function of Volume.</p>
<p>Write Surface Area, <code class='latex inline'>S</code>, as a function of Volume, <code class='latex inline'>V</code>. Also, write Volume, <code class='latex inline'>V</code>, as a function of Surface Area, <code class='latex inline'>S</code>.</p>
<p>The power <code class='latex inline'>4^3</code> means that 4 is multiplied by itself three times. Explain the meaning of <code class='latex inline'>4^{2.5}</code>.</p>
<p>Find the volume of a cube with surface area of </p> <ul> <li>i) <code class='latex inline'>294 m^2</code></li> <li>ii) <code class='latex inline'>36.8 m^2</code></li> </ul>
<p>Find the surface area of a cylindrical storage tank with height 10 m and volume <code class='latex inline'>1000 m^3</code>.</p>
<p>What is the side length of a cube with a volume of </p> <ul> <li>i) <code class='latex inline'>64 m^2</code>?</li> <li>ii) <code class='latex inline'>343 m^3</code>?</li> <li>iii) <code class='latex inline'>15.4 m^3</code>?</li> </ul>
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