Jetrva

<h3>Dobrodošli na Primjer Web Stranice</h3> \( \frac{ §§V1(-100,50,0.05)§§ }{4} \cdot \left(\frac{ §§V2(-101,-50,0.25)§§ }{6} + \frac{7}{ §§V3(1,10,1)§§ }\right) + §§V5(1,10,0.125)§§ + §§V5(-111,10,0.125)§§ \) <h5> Imena su : §§N0§§ - §§N1§§ - §§N2§§ </h5> Berechne \( 4 \times \left(\frac{1}{ §§V7(1,10,1)§§ } + \frac{ §§V6(1,10,1)§§ }{3}\right) \) Ovo je primjer teksta koji kombinira običan tekst s LaTeX notacijom za matematičke formule. \( \quad f(x)=\displaystyle\log{\frac{x^2-3x+2}{x+1}} \) Na primjer, možemo prikazati sljedeću formulu: Brzina svjetlosti u praznom prostoru definira se kao \(c = 299,792,458\) metara u sekundi. Možemo također prikazati kvadratni korijen formule: Kvadratni korijen iz broja \(x\) definira se kao \(\sqrt{x}\). <img src="http://gradiste.com/galerija/ljeto_jesen09/DSC06802.jpg" style="max-width: 500px;" alt="Image Description"> (1) The rectangular garden of §§N0§§ is §§V0(10,50,1)§§ m long and §§V1(5,30,1)§§ m wide. Calculate the perimeter and area of the garden. (2) §§Fm1§§ is cycling a distance of §§V2(3,15,0.5)§§ km at an average speed of §§V3(10,35,1)§§ km/h. How long will her ride take? (3) A parabola has the equation \( f(x) = §§V4(1,5,1)§§x^2 + §§V5(-10,10,1)§§x + §§V6(-20,20,1)§§ \). Find the coordinates of the vertex. (4) In a coordinate system, the points A(§§V7(-10,10,1)§§, §§V8(-10,10,1)§§) and B(§§V9(-10,10,1)§§, §§V10(-10,10,1)§§) are given. Calculate the length of the line segment AB. (5) The population of a city is growing exponentially at a rate of §§V11(1.01,1.10,0.01)§§ annually. How many people will live in the city after §§V12(5,20,1)§§ years, if the current population is §§V13(10000,50000,500)§§? (6) §§M2§§ buys a laptop for §§V14(500,1500,50)§§ €. Each year, the laptop loses §§V15(10,30,1)§§ % of its value. How much is the laptop worth after §§V16(1,5,1)§§ years? (7) A cone has a radius of §§V17(3,10,0.5)§§ cm and a height of §§V18(5,20,1)§§ cm. Calculate the volume of the cone. (8) §§Fm3§§ takes out a loan of §§V19(2000,10000,500)§§ € at an annual interest rate of §§V20(2,8,0.5)§§ %. How much will the debt be after §§V21(1,10,1)§§ years if no payments are made? (9) The linear function \( f(x) = §§V22(-5,5,1)§§x + §§V23(-10,10,1)§§ \) intersects the x-axis at which point? (10) The cube of §§N4§§ has an edge length of §§V24(1,10,1)§§ cm. Calculate the volume and surface area of the cube. (11) A figure is centrally dilated with center Z and dilation factor \( k = §§V25(-3,3,0.5)§§ \). Describe the effect of this dilation when \( k < 0 \) and when \( k > 0 \). (12) The point A has coordinates \( A(§§V26(-10,10,1)§§, §§V27(-10,10,1)§§) \). Calculate the image coordinates \( A' \) after a central dilation with center at the origin and factor \( k = §§V28(0.5,2,0.1)§§ \). (13) §§M5§§ constructs a central dilation of the figure with center Z and dilation factor \( k = §§V29(1,5,1)§§ \). The original figure has side lengths of §§V30(2,10,1)§§ cm. How long is the corresponding side in the dilated figure? (14) Two points B and B′ are centrally dilated with center Z(0,0). Point B has coordinates \( B(§§V31(-5,5,1)§§, §§V32(-5,5,1)§§) \), and B′ has coordinates \( B'(§§V33(-10,10,1)§§, §§V34(-10,10,1)§§) \). Calculate the dilation factor \( k \). (15) A figure is dilated with a negative dilation factor \( k = §§V35(-2,-0.5,0.1)§§ \). Explain the geometric meaning of this transformation regarding the orientation and position of the dilated figure. (16) A figure has corners \( A(§§V36(-8,8,1)§§, §§V37(-8,8,1)§§), B(§§V38(-8,8,1)§§, §§V39(-8,8,1)§§), C(§§V40(-8,8,1)§§, §§V41(-8,8,1)§§) \), and \( D(§§V42(-8,8,1)§§, §§V43(-8,8,1)§§) \). Calculate the image coordinates of these points after a central dilation with center \( Z(0,0) \) and dilation factor \( k = §§V44(1,2,0.1)§§ \). (17) §§Fm4§§ has a rectangular area with side lengths \( a = §§V45(4,10,1)§§ \) and \( b = §§V46(5,15,1)§§. She dilates the rectangle centrally with center \( Z \) and dilation factor \( k = §§V47(0.5,2,0.2)§§ \). Calculate the new side lengths and the area of the dilated rectangle. (18) A right-angled triangular figure has the corners \( A(§§V48(-10,10,1)§§, §§V49(-10,10,1)§§), B(§§V50(-10,10,1)§§, §§V51(-10,10,1)§§) \) and \( C(§§V52(-10,10,1)§§, §§V53(-10,10,1)§§) \). Calculate the image of the triangle after a central dilation with center \( Z(0,0) \) and dilation factor \( k = §§V54(1,3,0.5)§§ \). Also, determine the perimeter and area of the new triangle. (19) A parabola has the equation \( f(x) = §§V55(1,5,1)§§x^2 + §§V56(-10,10,1)§§x + §§V57(-5,5,1)§§ \). This parabola is centrally dilated with center \( Z(0,0) \) and dilation factor \( k = §§V58(0.5,2,0.1)§§ \). Calculate the new coordinates of the vertex and the new behavior of the parabola. (20) The points \( A(§§V59(-15,15,1)§§, §§V60(-15,15,1)§§) \) and \( B(§§V61(-15,15,1)§§, §§V62(-15,15,1)§§) \) are centrally dilated with the center at point \( Z(§§V63(-5,5,1)§§, §§V64(-5,5,1)§§) \) and dilation factor \( k = §§V65(0.1,5,0.1)§§ \). Calculate the dilation factor and the image coordinates of the points after dilation. <table class="table table-bordered table-striped"> <thead class="thead-dark"> <tr> <th>Term</th> <th>Explanation</th> </tr> </thead> <tbody> <tr> <td>Center \( Z \)</td> <td>The fixed point from which the dilation occurs. All image points are stretched relative to this point.</td> </tr> <tr> <td>Dilation factor \( k \)</td> <td>Indicates how much the figure is enlarged (\( k > 1 \)), reduced (\( 0 < k < 1 \)), reflected (\( k < 0 \)), or unchanged (\( k = 1 \)).</td> </tr> <tr> <td>Behavior of the Parabola</td> <td>The central dilation of a parabola changes its vertex and its width. If \( k > 1 \), the parabola becomes narrower; if \( 0 < k < 1 \), it becomes wider.</td> </tr> <tr> <td>Area after Dilation</td> <td>The area of the figure changes by the square of the dilation factor. If \( k \) is the dilation factor, the area changes by a factor of \( k^2 \).</td> </tr> <tr> <td>Perimeter after Dilation</td> <td>The perimeter of a figure changes linearly with the dilation factor \( k \). If the original perimeter is \( U_0 \), the new perimeter is \( U = k \cdot U_0 \).</td> </tr> </tbody> </table>